Ex.8.4 Q5 Introduction to Trigonometry Solution - NCERT Maths Class 10

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Question

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

(i) \(\begin{align} {{\text{(cosec}\,\text{ }\!\!\theta\!\!\text{ }-\text{cot}\,\text{ }\!\!\theta\!\!\text{ )}}^{\text{2}}}\text{=}\frac{\text{1}-\text{cos}\,\text{ }\!\!\theta\!\!\text{ }}{\text{1+cos}\,\text{ }\!\!\theta\!\!\text{ }}   \end{align}\)

(ii) \(\begin{align} \frac{\text{cos}\,A}{\text{1+sin}\,A}\,\,\text{+}\,\frac{\text{1+sin}\,A}{\text{cos}\,A}\text{=2sec}A  \end{align}\)

(iii) \(\begin{align} \frac{\text{tan}\,\text{ }\!\!\theta\!\!\text{ }}{\text{1}-\text{cot}\,\text{ }\!\!\theta\!\!\text{ }}\,\,\text{+}\,\,\frac{\text{cot}\,\text{ }\!\!\theta\!\!\text{ }}{\text{1}-\text{tan}\,\text{ }\!\!\theta\!\!\text{ }}\text{=1+sec}\,\text{ }\!\!\theta\!\!\text{ }\,\text{cosec}\,\text{ }\!\!\theta\!\!\text{ }   \end{align}\)

(iv) \(\begin{align} \frac{\text{1+sec}\; A}{\text{sec}\,A}\text{=}\frac{\text{si}{{\text{n}}^{\text{2}}}A}{\text{1}-\text{cos}\,A}   \end{align}\)

(v) \(\begin{align} \frac{\text{cos}\,\text{A}-\text{sin}\,\text{A+1}}{\text{cos}\,\text{A+sin}\,\text{A+1}}\text{=cosec}\,\text{A}\,\text{+}\,\text{cotA}  \end{align}\)

(vi) \(\begin{align}  \sqrt{\frac{\text{1+sinA}}{\text{1-sinA}}}\,\text{=}\,\text{sec}\,\text{A}\,\text{+}\,\text{tan}\,\text{A}  \end{align}\)

(vii) \(\begin{align}  \frac{\text{sin}\,\text{ }\!\!\theta\!\!\text{ }-\text{2si}{{\text{n}}^{\text{3}}}\text{ }\!\!\theta\!\!\text{ }}{\text{2cos}\,\text{ }\!\!\theta\!\!\text{ }-\text{cos}\,\text{ }\!\!\theta\!\!\text{ }}\,\text{=}\,\text{tan}\,\text{ }\!\!\theta\!\!\text{ }  \end{align}\)

(viii) \(\begin{align} {{\text{(sin} A\,\text{+}\,\text{cosec}\,A)}^{\text{2}}}\text{+(cos}\,A\,\text{+}\,\text{sec}\,A{{\text{)}}^{\text{2}}}\,\end{align}\)\(\begin{align} \text{=}\,\text{7+ta}{{\text{n}}^{\text{2}}}A\,\text{+}\,\text{co}{{\text{t}}^{\text{2}}}A  \end{align}\)

(ix) \(\begin{align} \text{(cosecA}-\text{sinA)}\,\text{(secA}-\text{cosA)}\,\text{=}\,\frac{\text{1}}{\,\text{tan}\,\text{A+cot}\,\text{A}} \end{align}\)

(x) \(\begin{align}  \left( \frac{1+{{\tan }^{2}}\text{A}}{1+{{\cot }^{2}}\text{A}} \right)={{\left( \frac{1-\tan \text{A}}{1-\cot \text{A}} \right)}^{2}} = \rm tan^2 A\\ \end{align}\) 

Text Solution

 

Reasoning:

\(\begin{align}{\sin ^{2} A+\cos ^{2} A=1} \\ {\text{cosec} ^{2} A=1+\cot ^{2} A} \\ {\sec ^{2} A=1+\tan ^{2} A}\end{align}\)

Steps:

(i) \(\begin{align}{(\text{cosec}\; \theta-\cot \theta)^{2}}&={\frac{1-\cos \theta}{1+\cos \theta}} \\ \end{align}\)
\(\begin{align} {\text { L.H.S }}&={(\text{cosec}\; \theta-\cot \theta)^{2}} \qquad\ldots \dots \ldots\end{align}\) \((1)\)

We know that the trigonometric functions,

\(\begin{align} \cot \,(x)&=\frac{\cos \,(x)}{\sin \,(x)} =\frac{1}{\tan \,(x)} \\ \operatorname{cosec}\,(x) &=\frac{1}{\sin \,(x)} \end{align}\)

By substituting the above function in Equation (1)

\(\begin{align} {\rm (cosec \theta - cot \theta)^2} &\text{=}\,\,{{\left( \frac{\text{1}}{\text{sin }\!\!\theta\!\!\text{ }}-\frac{\text{cos}\,\text{ }\!\!\theta\!\!\text{ }}{\text{sin}\,\text{ }\!\!\theta\!\!\text{ }} \right)}^{\text{2}}}\end{align}\)

\(\begin{align} \text{=}\,\frac{{{\text{(1}-\text{cos}\,\text{ }\!\!\theta\!\!\text{ )}}^{\text{2}}}}{{{\text{(sin}\,\text{ }\!\!\theta\!\!\text{ )}}^{\text{2}}}}\,\,\,\,\,\,\end{align}\)

\(\begin{align}  \text{=}\,\frac{{{\text{(1}-\text{cos}\,\text{ }\!\!\theta\!\!\text{ )}}^{\text{2}}}}{\text{si}{{\text{n}}^{\text{2}}}\,\text{ }\!\!\theta\!\!\text{ }}   \end{align}\)

\(\begin{align} \text{=}\,\frac{{{\text{(1}-\text{cos}\,\text{ }\!\!\theta\!\!\text{ )}}^{\text{2}}}}{\text{1}-\text{co}{{\text{s}}^{\text{2}}}\text{ }\!\!\theta\!\!\text{ }}\,\,\,\,\,\,\,\,   \end{align}\) (By identity \(\sin^2 A \) \(+\) \(\cos^2 A =\) \(1\) Hence, \(1-\cos^2 A\)\( = \sin^2 A\))

\(\begin{align} \text{=}\,\frac{{{\text{(1}-\text{cos}\,\text{ }\!\!\theta\!\!\text{ )}}^{\text{2}}}}{\text{(1}-\text{cos}\,\text{ }\!\!\theta\!\!\text{ )(1+cos}\,\text{ }\!\!\theta\!\!\text{ )}}   \end{align}\) [Using \(a^2 - b^2\)\(= \)\((a+b)(a-b)\)]

\(\begin{align} \text{=}\,\frac{\text{1}-\text{cos}\,\text{ }\!\!\theta\!\!\text{ }}{\text{1+cos}\,\text{ }\!\!\theta\!\!\text{ }}  \end{align}\)

=R.H.S.

(ii) \(\begin{align} \frac{\text{cos}A}{\text{1+sin}A}&+\frac{\text{1+sin}A}{\text{cos}A}\,\,=\,\text{2sec}A   \end{align}\)

LHS\(\begin{align}  =\,\frac{\text{cosA}}{\text{1+sinA}}\text{+}\frac{\text{1+sinA}}{\text{cosA}}  \end{align}\)

\(\begin{align}   \text{=}\,\frac{\text{co}{{\text{s}}^{\text{2}}}\text{A+(1+sinA}{{\text{)}}^{\text{2}}}}{\text{(1+sinA)}\,\text{(cosA)}} \end{align}\) 

\(\begin{align}    \text{=}\,\,\frac{\text{co}{{\text{s}}^{\text{2}}}A\text{+1+si}{{\text{n}}^{\text{2}}}A\text{+2sin}A}{\text{(1+sin}A)\,\text{(cos}A)}  \end{align}\)

\(\begin{align}     \text{=}\frac{\text{si}{{\text{n}}^{\text{2}}}A\text{+co}{{\text{s}}^{\text{2}}}A\text{+1+2sin}A}{\text{(1+sin}A)\text{(cos}A)}  \end{align}\)

\(\begin{align}     \text{=}\frac{\text{1+1+2sin}A}{\text{(1+sin}A)\text{(cos}A)}  \end{align}\) (By identify \(\sin^2 A + \cos ^2 A=1\))

\(\begin{align}      \text{=}\frac{\text{2+2sin}A}{\text{(1+sin}A)\text{(cos}A)}  \end{align}\)

\(\begin{align}       \text{=}\frac{\text{2(1+sin}A)}{\text{(1+sin}A)\text{(cos}A)}\,\,\,\,\,\,\,\,\,\,   \end{align}\) (By taking \(2\) common and simplifying)

\(\begin{align}        \text{=}\frac{\text{2}}{\text{cos}A} \end{align}\)

\(\begin{align}   \text{=2sec}A  \end{align}\)

=R.H.S.

(iii) \(\begin{align}{ \frac{\tan \theta}{1-\cot \theta}+\frac{\cot \theta}{1-\tan \theta}=1+\sec \theta \; \text{cosec}\; \theta} \\ \end{align} \)

LHS\(\begin{align}  {=\frac{\tan \theta}{1-\cot \theta}+\frac{\cot \theta}{1-\tan \theta}} \qquad\qquad \dots \dots(1) \end{align}\)

We know that the trigonometric functions,

\(\begin{align} & \tan \,(x)=\frac{\sin (x)}{\cos (x)} \\ & \cot \,(x)=\frac{\cos (x)}{\sin (x)}=\frac{1}{\tan (x)} \end{align}\)

By substituting the above relations in Equation (1),

\(\begin{align}   =\frac{\frac{\sin \theta }{\cos \theta }}{1-\frac{\cos \theta }{\sin \theta }}+\frac{\frac{\cos \theta }{\sin \theta }}{1-\frac{\cos \theta }{\sin \theta }}  \end{align}\)

\(\begin{align}    =\frac{\frac{\sin \theta }{\cos \theta }}{\frac{\sin \theta -\cos \theta }{\sin \theta }}+\frac{\frac{\cos \theta }{\sin \theta }}{\frac{\cos \theta -\sin \theta }{\cos \theta }}  \end{align}\) 

\(\begin{align}   =\frac{{{\sin }^{2}}\theta }{\cos \theta (\sin \theta -\cos \theta )}+\frac{{{\cos }^{2}}\theta }{\sin \theta (\sin \theta -\cos \theta )} \end{align}\)

Taking \(\begin{align}    \,\,\frac{\text{1}}{\text{(sin }\!\!\theta\!\!\text{ }-\text{cos }\!\!\theta\!\!\text{ )}} \end{align}\) as common

\(\begin{align}   \text{=}\,\,\frac{\text{1}}{\text{(sin }\!\!\theta\!\!\text{ }-\text{cos }\!\!\theta\!\!\text{ )}}\left[ \frac{\text{si}{{\text{n}}^{\text{2}}}\text{ }\!\!\theta\!\!\text{ }}{\text{cos }\!\!\theta\!\!\text{ }}-\frac{\text{co}{{\text{s}}^{\text{2}}}\text{ }\!\!\theta\!\!\text{ }}{\text{sin }\!\!\theta\!\!\text{ }} \right] \\ \text{=}\,\,\frac{\text{1}}{\text{(sin }\!\!\theta\!\!\text{ }-\text{cos }\!\!\theta\!\!\text{ )}}\left[ \frac{\text{si}{{\text{n}}^{\text{3}}}\text{ }\!\!\theta\!\!\text{ }-\text{co}{{\text{s}}^{\text{3}}}\text{ }\!\!\theta\!\!\text{ }}{\text{sin}\,\text{ }\!\!\theta\!\!\text{ }\,\text{cos }\!\!\theta\!\!\text{ }} \right] \end{align}\)

Using \(\begin{align}  \,\,{{\text{a}}^{\text{3}}}\text{-}{{\text{b}}^{\text{3}}}\text{=(a-b)}\left( {{\text{a}}^{\text{2}}}\text{+}\,\text{ab}\,\text{+}\,{{\text{b}}^{\text{2}}} \right)  \end{align}\)

\(\begin{align}  \text{=}\frac{\text{1}}{\text{(sin}\,\text{ }\!\!\theta\!\!\text{ }-\text{cos}\,\text{ }\!\!\theta\!\!\text{ )}} \end{align}\) \(\begin{align}\left[ \frac{\text{(sin}\,\text{ }\!\!\theta\!\!\text{ }-\text{cos}\,\text{ }\!\!\theta\!\!\text{ )}\left( \text{si}{{\text{n}}^{\text{2}}}\text{ }\!\!\theta\!\!\text{ }\,\text{+}\,\text{co}{{\text{s}}^{\text{2}}}\text{ }\!\!\theta\!\!\text{ }\,\,\text{+}\,\,\text{sin}\,\text{ }\!\!\theta\!\!\text{ }\,\text{cos }\!\!\theta\!\!\text{ } \right)}{\text{sin }\!\!\theta\!\!\text{ }\,\text{cos }\!\!\theta\!\!\text{ }} \right]  \end{align}\)

\(\begin{align}\text{=}\frac{\text{(1}\,\text{+}\,\text{sin}\,\text{ }\!\!\theta\!\!\text{ }\,\text{cos}\,\text{ }\!\!\theta\!\!\text{ )}}{\text{(sin}\,\text{ }\!\!\theta\!\!\text{ cos}\,\text{ }\!\!\theta\!\!\text{ )}}\quad\end{align}\) (By identity \(\sin^2 A \) \(+ \) \(\cos^2 A \) \(= 1\) )

\(\begin{align}    \text{=1+}\,\text{sec}\,\text{ }\!\!\theta\!\!\text{ }\,\text{cosec}\,\text{ }\!\!\theta\!\!\text{ } \\ \end{align}\) 

=R.H.S.

(iv) \(\begin{align} \frac{1+\sec A}{\sec A}&=\frac{\sin ^{2} A}{1-\cos A} \\ \end{align}\)

LHS \(\begin{align}=\frac{1+\sec A}{\sec A}\qquad \qquad \dots \dots (1) \end{align}\)

We know that the trigonometric functions,

\(\begin{align}\sec \,(x)=\frac{1}{\cos \,(x)}\end{align}\)

By substituting the above function in Equation (1),

\[\begin{align}
\frac{{{\rm{1}}\,{\rm{ + }}\,{\rm{sec}}\,{\rm{A}}}}{{{\rm{sec}}\,{\rm{A}}}}{\rm{ = }}\,\frac{{{\rm{1 + }}\frac{{\rm{1}}}{{{\rm{cosA}}}}}}{{\frac{{\rm{1}}}{{{\rm{cosA}}}}}}\\
{\rm{ = }}\,\frac{{\frac{{{\rm{cosA + 1}}}}{{{\rm{cosA}}}}}}{{\frac{{\rm{1}}}{{{\rm{cosA}}}}}}\\
 = \frac{{{\rm{cosA + 1}}}}{{{\rm{cosA}}}} \times \frac{{{\rm{cosA}}}}{1}\\
{\rm{ = }}\,{\rm{(1 + cosA)}}
\end{align}\]

By multiplying \(\text{( 1 - cos A)}\) in both denominator and numerator

\(\begin{align} &\Rightarrow\frac{(1-\cos A)(1+\cos A)}{(1-\cos A)} \\   &=\frac{1-\cos ^{2} A}{1-\cos A} \\ &=\frac{\sin ^{2} A}{1-\cos A} \qquad \text { [By Identity } \sin ^{2} A+\cos ^{2} A =1] \end{align}\)

=R.H.S.

(v) \(\begin{align} {\frac{\cos A-\sin A+1}{\cos A+\sin A-1}}&={\text{cosec} A+\cot A} \end{align}\)

\({\rm{L}}.{\rm{H}}.{\rm{S}} = \frac{{\cos {\rm{A}} - \sin {\rm{A}} + 1}}{{\cos {\rm{A}} + \sin {\rm{A}} - 1}}\)

Diving both numerator and denominator by \(\text{sin A}\)

\(\begin{align}\text{=}\frac{\frac{\text{cos}\,\text{A}}{\text{sin}\,\text{A}}-\frac{\text{sin}\,\text{A}}{\text{sin}\,\text{A}}\text{+}\frac{\text{1}}{\text{sinA}}}{\frac{\text{cos}\,\text{A}}{\text{sin}\,\text{A}}\text{+}\frac{\text{sin}\,\text{A}}{\text{sin}\,\text{A}}\text{+}\frac{\text{1}}{\text{sin}\,\text{A}}}\end{align}\)

We know that the trigonometric functions,

\(\,\begin{align} \text{cot}\,\text{(x)=}\frac{\text{cos}\,\text{(x)}}{\text{sin}\,\text{(x)}}\,\,\,&=\frac{\text{1}}{\text{tan}\,\text{(x)}} \\ \text{cosec}\,\text{(x)}&=\frac{\text{1}}{\text{sin}\,\text{(x)}} \\ \text{We}\,\text{get} \\ \,\,\,&\Rightarrow =\frac{\text{cot}\,\text{A - 1+}\,\,\text{cosec}\,\text{A}}{\text{cot}\,\text{A}\,\text{+ 1- cosec}\,\text{A}} \\ &\Rightarrow =\frac{\text{cot}\,\text{A - (1-}\,\,\text{cosec}\,\text{A)}}{\text{cot}\,\text{A}\,\text{+ (1- cosec}\,\text{A)}}\end{align}\)

We know that, \(\,\begin{align}\text{1 + co}{{\text{t}}^{\text{2}}}\text{A = Cose}{{\text{c}}^{\text{2}}}\text{A}\end{align}\)

Hence multiplying \([\cot A – (1 – \rm{cosec}\; A)]\) in numerator and denominator

\(\begin{align} & \text{=}\frac{\text{ }\!\!\{\!\!\text{ (cot}\,\text{A)}-\text{(1}-\text{cosec}\,\text{A) }\!\!\}\!\!\text{ }\!\!\{\!\!\text{ (cot}\,\text{A)}-\text{(1-cosec}\,\text{A) }\!\!\}\!\!\text{ }}{\text{ }\!\!\{\!\!\text{ (cot}\,\text{A)}\,\text{+}\,\text{(1}-\text{cosec}\,\text{A) }\!\!\}\!\!\text{ }\!\!\{\!\!\text{ (cot}\,\text{A)}-\text{(1}-\text{cosec}\,\text{A) }\!\!\}\!\!\text{ }} \\ & \text{=}\frac{{{\text{(cot}\,\text{A}-\text{1}\,\,\text{+}\,\,\text{cosec}\,\text{A)}}^{\text{2}}}}{{{\text{(cot}\,\text{A)}}^{\text{2}}}-{{\text{(1}-\text{cosecA)}}^{\text{2}}}} \\ & \text{=}\frac{\text{co}{{\text{t}}^{\text{2}}}\text{A}\,\text{+}\,\text{1+}\,\text{cose}{{\text{c}}^{\text{2}}}\text{A}-\text{2}\,\text{cot}\,\text{A}-\text{2cosec}\,\text{A}\,\text{+}\,\text{2}\,\text{cot}\,\text{A}\,\text{cosecA}}{\text{co}{{\text{t}}^{\text{2}}}\text{A}-\left( \text{1+cose}{{\text{c}}^{\text{2}}}\text{A}-\text{2cosec}\,\text{A} \right)} \\ & \text{=}\frac{\text{2}\,\text{cose}{{\text{c}}^{\text{2}}}\text{A}\,\text{+}\,\text{2cot}\,\text{A}\,\,\text{cosecA}-\text{2cot}\,\text{A}-\text{2cosecA}}{\text{co}{{\text{t}}^{\text{2}}}\text{A}-\text{1}\,\text{+}\,\text{cose}{{\text{c}}^{\text{2}}}\text{A}\,\text{+}\,\text{2}\,\text{cosecA}}\,\, \\ & \text{=}\frac{\text{2cosec}\,\text{A}\,\text{(cosec}\,\text{A}\,\text{+}\,\text{cot}\,\text{A)}\,-\text{2(cot}\,\text{A}-\text{cosec}\,\text{A)}}{\text{co}{{\text{t}}^{\text{2}}}\text{A}-\,\text{cose}{{\text{c}}^{\text{2}}}\text{A}-\text{1}\,\text{+}\,\text{2}\,\text{cosec}\,\text{A}} \\ & \text{=}\frac{\text{(cosec}\,\text{A}\,\text{+}\,\text{cotA)}\,\text{(2}\,\text{cosecA}-\text{2)}}{\text{1}-\text{1+}\,\text{2cosec}\,\text{A}} \\ & \text{=}\frac{\text{(cosecA}\,\text{+}\,\text{cotA)}\,\text{(2}\,\text{cosecA}\,-\text{2)}}{\text{(2}\,\text{cosec}\,\text{A}-\text{2)}} \\ & =\text{cosec A + cot A} \\ & =\text{R}\text{.H}\text{.S} \\ \end{align}\)

(vi) \(\begin{align} \sqrt{\frac{1+\sin A}{1-\sin A}}&=\sec A+\tan A \\\end{align}\)

LHS \(\begin{align}  =\sqrt{\frac{1+\sin A}{1-\sin A}}\qquad \qquad\dots \dots (1) \end{align}\)

Multiplying and dividing by \(\begin{align}\sqrt{\left( \text{1+}\,\text{sin}\,\text{A} \right)}\end{align}\)

\[\begin{array}{l}
 \Rightarrow \sqrt {\frac{{{\rm{(1 + }}\,{\rm{sin}}\,{\rm{A)(1 + }}\,{\rm{sin}}\,{\rm{A)}}}}{{{\rm{(1}} - {\rm{sin}}\,{\rm{A)(1 + }}\,{\rm{sin}}\,{\rm{A)}}}}} \\
 = \sqrt {\frac{{{{{\rm{(1 + }}\,{\rm{sin}}\,{\rm{A)}}}^2}}}{{{\rm{(1}} - {\rm{si}}{{\rm{n}}^2}{\rm{A)}}}}} {\rm{                 }}\qquad\left[ {{a^2} - {b^2} = \left( {a-b} \right)\left( {a + b} \right),} \right]\\
\,\begin{array}{*{20}{l}}
\begin{array}{l}
{\rm{ = }}\frac{{{\rm{(1 + }}\,{\rm{sinA)}}}}{{\sqrt {{\rm{1}} - {\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{A}}} }}\,\\
{\rm{ = }}\,\frac{{{\rm{1 + sin}}\,{\rm{A}}}}{{\sqrt {{\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{A}}} }}
\end{array}\\
\begin{array}{l}
{\rm{ = }}\frac{{{\rm{1 + }}\,{\rm{sin}}\,{\rm{A}}}}{{{\rm{cos}}\,{\rm{A}}}}\\
 = \frac{1}{{{\rm{cos}}\,{\rm{A}}}}\,\, + \frac{{{\rm{sin}}\,{\rm{A}}}}{{{\rm{cos}}\,{\rm{A}}}}\\
{\rm{ = }}\,\,{\rm{sec}}\,{\rm{A}}\,{\rm{ + }}\,{\rm{tan}}\,{\rm{A}}
\end{array}\\
{{\rm{ = R}}{\rm{.H}}{\rm{.S}}}
\end{array}
\end{array}\]

 

(vii) \(\begin{align}  { \frac{\sin \theta-2 \sin ^{3} \theta}{2 \cos \theta-\cos \theta}}&={\tan \theta} \end{align}\) 

LHS \(\begin{align}  &={\frac{\sin \theta-2 \sin ^{3} \theta}{2 \cos ^{3} \theta-\cos \theta}}\end{align}\)

Taking \(\text{Sin }\!\!\;\theta\!\!\;\text{ and Cos }\!\!\;\theta\!\!\text{ }\) common in both numerator and denominator respectively.

\(\begin{align}=\Rightarrow \frac{\sin \text{ }\!\!\theta\!\!\text{ }\left( 1-2{{\sin }^{2}}\text{ }\!\!\theta\!\!\text{ } \right)}{\cos \text{ }\!\!\theta\!\!\text{ }\left( 2{{\cos }^{2}}\text{ }\!\!\theta\!\!\text{ }-1 \right)}\end{align}\)

By Identity \(\text{si}{{\text{n}}^{\text{2}}}\text{A + co}{{\text{s}}^{\text{2}}}\text{A = 1}\) hence, \(\text{co}{{\text{s}}^{\text{2}}}\text{A = 1 - si}{{\text{n}}^{\text{2}}}\text{A}\) and substituting this in the above equation,

\[\begin{array}{l}
 \Rightarrow \frac{{{\rm{sin}}\,{\rm{\theta }}\left( {{\rm{1}} - {\rm{2si}}{{\rm{n}}^{\rm{2}}}{\rm{\theta }}} \right)}}{{{\rm{cos}}\,{\rm{\theta }}\left\{ {{\rm{2}}\left( {{\rm{1}} - {\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{\theta }}} \right) - {\rm{1}}} \right\}}}\\
 = \frac{{{\rm{sin}}\,{\rm{\theta }}\left( {{\rm{1}} - {\rm{2si}}{{\rm{n}}^{\rm{2}}}{\rm{\theta }}} \right)}}{{{\rm{cos}}\,{\rm{\theta }}\left( {{\rm{2}} - 2{\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{\theta }} - {\rm{1}}} \right)}}\\
{\rm{ = }}\,\frac{{{\rm{sin}}\,{\rm{\theta }}\left( {{\rm{1}} - {\rm{2si}}{{\rm{n}}^{\rm{2}}}{\rm{\theta }}} \right)}}{{{\rm{cos}}\,{\rm{\theta }}\left( {{\rm{1}} - {\rm{2si}}{{\rm{n}}^{\rm{2}}}{\rm{\theta }}} \right)}}\\
{\rm{ = }}\,\frac{{{\rm{sin}}\,{\rm{\theta }}}}{{{\rm{cos}}\,{\rm{\theta }}}}\,\\
{\rm{ = }}\,{\rm{tan}}\,{\rm{\theta }}\\
{\rm{ = RHS}}
\end{array}\]

(viii) \(\begin{align} {(\sin \mathrm{A}+\text{cosec}\; \mathrm{A})^{2}+(\cos \mathrm{A}+\sec \mathrm{A})^{2}}\end{align}\)\(\begin{align}  ={7+\tan ^{2} \mathrm{A}+\cot ^{2} \mathrm{A}}  \end{align}\)

LHS \(\begin{align} ={(\sin \mathrm{A}+\text{cosec} \mathrm{A})^{2}+(\cos \mathrm{A}+\sec \mathrm{A})^{2}}\end{align}\)

By using\(\begin{align}{{\left( \text{a + b } \right)}^{\text{2}}}&\text{ = }{{\text{a}}^{\text{2}}}\text{ + 2ab +}\,{{\text{b}}^{\text{2}}}\end{align}\)

\(\begin{align} \Rightarrow &\text{=}\,\text{si}{{\text{n}}^{\text{2}}}\text{A}\,\text{+}\,\text{cose}{{\text{c}}^{\text{2}}}\text{A}\,\text{+}\,\text{2}\,\text{sin}\,\text{A}\,\,\text{cosec}\,\text{A}\,   \end{align}\)\(\begin{align} \text{+}\,\text{co}{{\text{s}}^{\text{2}}}\text{A}\,\text{+}\,\text{se}{{\text{c}}^{\text{2}}}\text{A}\,\text{+}\,\text{2}\,\text{cosA}\,\,\,\text{secA}   \end{align}\)

By rearranging and using \(\begin{align}\text{sec A = }\frac{1}{\text{cos A}} \rm and \;\rm cosec A = \frac{{1}}{sin A}\end{align}\)

\(\begin{align} \Rightarrow \text{}\,\left( \text{si}{{\text{n}}^{\text{2}}}\text{A}\,\text{+}\,\text{co}{{\text{s}}^{\text{2}}}\text{A} \right)\text{+}\left( \text{cose}{{\text{c}}^{\text{2}}}\text{A}\,\text{+}\,\text{se}{{\text{c}}^{\text{2}}}\text{A} \right)\,   \end{align}\)\(\begin{align}\text{+}\,\text{2}\,\text{sin}\,\text{A}\left( \frac{\text{1}}{\text{sin}\,\text{A}} \right)\,\text{+}\,\text{2cos}\,\text{A}\left( \frac{\text{1}}{\text{cos}\,\text{A}} \right)\end{align}\) 

Hence \(\left( {{\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{A}}\,{\rm{ + }}\,{\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{A}}} \right)\,{\rm{ = }}\,{\rm{1, cose}}{{\rm{c}}^{\rm{2}}}{\rm{A}}\,{\rm{ = }}\left( {{\rm{1 + co}}{{\rm{t}}^{\rm{2}}}{\rm{A}}} \right){\rm{ and }}\left( {{\rm{se}}{{\rm{c}}^{\rm{2}}}{\rm{A}}\, - {\rm{ta}}{{\rm{n}}^{\rm{2}}}{\rm{A}}} \right)\,{\rm{ = 1}}\)

\(\begin{align} & =(1)+\left( 1+{{\cot }^{2}}\text{A}+1+{{\tan }^{2}}\text{A} \right)+(2)+(2) \\ & =7+{{\tan }^{2}}\text{A}+{{\cot }^{2}}\text{A} \\   \end{align}\)

=R.H.S.

(ix) \(\begin{align} (\text{cosec}\,\text{A}-\sin \text{A})\,(\sec \text{A}-\cos \text{A})&=\frac{1}{\tan \text{A}+\cot \text{A}}  \end{align}\)

LHS \(\begin{align} =(\operatorname{cosec}\text{A}-\sin \text{A})\,(\sec \text{A}-\cos \text{A})\ldots \ldots \ldots \text{ (1)} \\ \end{align}\)

We know that the trigonometric functions,

\(\begin{align} & \sec \,(x)=\frac{1}{\cos \,(x)} \\ & \operatorname{cosec}\,(x)=\frac{1}{\sin \,(x)} \end{align}\)

By substituting the above relations in Equation (1)

\[\begin{array}{l}
 \Rightarrow \,\left( {\frac{{\rm{1}}}{{{\rm{sin}}\,{\rm{A}}}} - {\rm{sin}}\,{\rm{A}}} \right)\left( {\frac{{\rm{1}}}{{{\rm{cosA}}}} - {\rm{cos}}\,{\rm{A}}} \right)\\
{\rm{ = }}\,\left( {\frac{{{\rm{1}} - {\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{A}}}}{{{\rm{sin}}\,{\rm{A}}}}} \right)\left( {\frac{{{\rm{1}} - {\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{A}}}}{{{\rm{cos}}\,{\rm{A}}}}} \right)\\
{\rm{ = }}\,\frac{{{\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{A si}}{{\rm{n}}^{\rm{2}}}{\rm{A}}}}{{{\rm{sin}}\,{\rm{A cos}}\,{\rm{A}}}}\\
{\rm{ = }}\,\frac{{{\rm{sin}}\,{\rm{A cos}}\,{\rm{A}}}}{1}\\
 = \frac{{{\rm{sin}}\,{\rm{A cos}}\,{\rm{A}}}}{{{\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{A + co}}{{\rm{s}}^{\rm{2}}}{\rm{A}}}}{\rm{      }}\left[ {\left( {{\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{A}}\,{\rm{ + }}\,{\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{A}}} \right)\,{\rm{ = }}\,{\rm{1}}} \right]\\
 = \frac{{\rm{1}}}{{\frac{{{\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{A + co}}{{\rm{s}}^{\rm{2}}}{\rm{A}}}}{{{\rm{sin}}\,{\rm{A cos}}\,{\rm{A}}}}}}{\rm{     }}\left[ {{\text{Dividing numerator and denominator by }}\left( {{\rm{sin}}\,{\rm{A cos}}\,{\rm{A}}} \right)} \right]\\
 = \frac{{\rm{1}}}{{\frac{{{\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{A}}}}{{{\rm{sin}}\,{\rm{A cos}}\,{\rm{A}}}} + \frac{{{\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{A}}}}{{{\rm{sin}}\,{\rm{A cos}}\,{\rm{A}}}}}}\\
 = \frac{{\rm{1}}}{{\frac{{{\rm{sin A}}}}{{{\rm{cos}}\,{\rm{A}}}} + \frac{{{\rm{cos A}}}}{{{\rm{sin}}\,{\rm{A}}}}}}\\
 = \frac{1}{{\tan {\rm{A}} + \cot {\rm{A}}}}\\
 = {\rm{RHS}}
\end{array}\]

(x) \(\begin{align}\,\,\left( \frac{1+{{\tan }^{2}}\text{A}}{1+{{\cot }^{2}}\text{A}} \right)={{\left( \frac{1-\tan \text{A}}{1-\cot \text{A}} \right)}^{2}}\end{align}\)

Taking LHS, \(\begin{align}\left( \frac{1+{{\tan }^{2}}A}{1+{{\cot }^{2}}A} \right)\end{align}\)

\(\begin{align} & \text{=}\,\frac{\text{se}{{\text{c}}^{\text{2}}}\text{A}}{\text{cose}{{\text{c}}^{\text{2}}}\text{A}} \\ & \text{=}\,\frac{\text{se}{{\text{c}}^{\text{2}}}\text{A}}{\text{cose}{{\text{c}}^{\text{2}}}\text{A}} \\ & \text{=}\,\frac{\frac{\text{1}}{\text{co}{{\text{s}}^{\text{2}}}\text{A}}}{\frac{\text{1}}{\text{si}{{\text{n}}^{\text{2}}}\text{A}}} \\ & \text{=}\,\frac{\text{1}}{\text{co}{{\text{s}}^{\text{2}}}\text{A}}\,\text{ }\!\!\times\!\!\text{ }\,\text{si}{{\text{n}}^{\text{2}}}\text{A} \\ & \text{=}\,\text{ta}{{\text{n}}^{\text{2}}}\text{A} \\&= RHS\\ \end{align}\)

Taking , \(\begin{align}{{\left( \frac{1-\tan \text{A}}{1-\cot \text{A}} \right)}^{2}}\end{align}\)

\(\begin{array}{c}
{\rm{ = }}\,{\left( {\frac{{{\rm{1}} - {\rm{tanA}}}}{{{\rm{1}} - \frac{{\rm{1}}}{{{\rm{tanA}}}}}}} \right)^{\rm{2}}}\\
{\rm{ = }}\,{\left( {\frac{{{\rm{1}} - {\rm{tanA}}}}{{\frac{{{\rm{tanA}} - 1}}{{\tan {\rm{A}}}}}}} \right)^{\rm{2}}}\\
 = {\left( {\left( {{\rm{1}} - {\rm{tanA}}} \right) \times \frac{{{\rm{tanA}}}}{{{\rm{tanA}} - 1}}} \right)^{\rm{2}}}\\
{\rm{ = }}\,{{\rm{(}} - {\rm{tanA)}}^{\rm{2}}}\\
{\rm{ = }}\,{\rm{ta}}{{\rm{n}}^{\rm{2}}}{\rm{A}}\\
{\rm{ = RHS}}
\end{array}\)

Hence, L.H.S = R.H.S.

  
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