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Trigonometric Identities

In this mini-lesson, we will explore trigonometric identities. We have already learned the primary trigonometric ratios sin, cos, and tan. We have also learned that there are three other trigonometric ratios sec, cosec, and cot which are the reciprocal of sin, cos, and tan respectively. How are these trigonometric ratios (sin, cos, tan, sec, cosec, and cot) connected with each other? They are connected through trigonometric identities (or in short trig identities). Let's start learning about trigonometric identities.

1.	What are Trigonometric Identities?
2.	Complementary and Supplementary Trigonometric Identities
3.	Sum and Difference Trigonometric Identities
4.	Double and Half Angles
5.	Sine and Cosine Rule
6.	Solved Examples on Trigonometric Identities
7.	Practice Questions on Trigonometric Identities
8.	FAQs on Trigonometric Identities

What are Trigonometric Identities?

Trigonometric identities are equations that relate different trigonometric functions and are true for any value of the variable that is there in the domain. Basically, an identity is an equation that holds true for all the values of the variable(s) present in it.

For example, some of the algebraic identities are:
(a+b)²= a²+ 2ab + b²
(a-b)²= a²- 2ab+ b²
(a+b)(a-b)= a²- b²

The algebraic identities relate just the variables whereas the trig identities relate the 6 trigonometric functions sine, cosine, tangent, cosecant, secant, and cotangent. Let's learn about each type of identities in detail.

Reciprocal Trigonometric Identities

We already know that the reciprocals of sin, cosine, and tangent are cosecant, secant, and cotangent respectively.

Thus, the reciprocal identities are:

sin θ = 1/cosecθ (OR) cosec θ = 1/sinθ
cos θ = 1/secθ (OR) sec θ = 1/cosθ
tan θ = 1/cotθ (OR) cot θ = 1/tanθ

Pythagorean Trigonometric Identities

The Pythagorean trigonometric identities in trigonometry are derived from the Pythagoras theorem. Applying Pythagoras theorem to the right-angled triangle below, we get:

Opposite²+ Adjacent²= Hypotenuse²

Dividing both sides by Hypotenuse²

Opposite²/Hypotenuse² + Adjacent²/Hypotenuse² = Hypotenuse²/Hypotenuse²

sin²θ + cos²θ = 1

This is one of the Pythagorean identities. In the same way, we can derive two other Pythagorean trigonometric identities.

1+tan²θ = sec²θ
1+cot²θ = cosec²θ

Complementary and Supplementary Trigonometric Identities

The complementary angles are a pair of two angles such that their sum is equal to 90°. The complement of an angle θ is (90 - θ). The trigonometric ratios of complementary angles are:

sin (90°- θ) = cos θ
cos (90°- θ) = sin θ
cosec (90°- θ) = sec θ
sec (90°- θ) = cosec θ
tan (90°- θ) = cot θ
cot (90°- θ) = tan θ

The supplementary angles are a pair of two angles such that their sum is equal to 180°. The supplement of an angle θ is (180 - θ). The trigonometric ratios of supplementary angles are:

sin (180°- θ) = sinθ
cos (180°- θ) = -cos θ
cosec (180°- θ) = cosec θ
sec (180°- θ)= -sec θ
tan (180°- θ) = -tan θ
cot (180°- θ) = -cot θ

Sum and Difference Trigonometric Identites

The sum and difference identities include the formulas of sin(A+B), cos(A-B), cot(A+B), etc.

sin (A+B) = sin A cos B + cos A sin B
sin (A-B) = sin A cos B - cos A sin B
cos (A+B) = cos A cos B - sin A sin B
cos (A-B) = cos A cos B +sin A sin B
tan (A+B) = (tan A + tan B)/ (1 - tan A tan B)
tan (A-B) = (tan A - tan B)/ (1 + tan A tan B)

Double and Half Angles Trigonometric Identites

Double angle formulas: The double angle trigonometric identities can be obtained by using the sum and difference formulas.

For example, from the above formulas:

sin (A+B) = sin A cos B + cos A sin B

Substitute A = B = θ on both sides here, we get:

sin (θ + θ) = sinθ cosθ + cosθ sinθ
sin 2θ = 2 sinθ cosθ

In the same way, we can derive the other double angle identities.

sin 2θ = 2 sinθ cosθ
cos 2θ = cos²θ - sin²θ
= 2 cos²θ - 1
= 1 - 2 sin²θ
tan 2θ = (2 tanθ)/ (1 - tan²θ)

Half Angle Formulas

Using one of the above double angle formulas,
cos 2θ = 1 - 2 sin²θ
2 sin²θ = 1- cos 2θ
sin²θ = (1-cos2θ)/(2)
sin θ = \(\pm \sqrt{\dfrac{1-cos 2θ}{2}}\)

Replacing θ by θ/2 on both sides,

\[sin \frac{θ}{2}=\pm \sqrt{\frac{1-cos θ}{2}}\]

This is the half-angle formula of sin.

In the same way, we can derive the other half-angle formulas.

\[\begin{array}{l}
1.\,\,sin \dfrac{θ}{2}=\pm \sqrt{\dfrac{1-cos θ}{2}} \\[0.2cm]
2. \,\, cos \dfrac{θ}{2}=\pm \sqrt{\dfrac{1+cos θ}{2}} \\[0.2cm]
3. \,\,\tan \dfrac{θ}{2}=\pm \sqrt{\dfrac{1-cos θ}{1+cos θ}}
\end{array}\]

The trigonometric identities that we have learned are derived using the right-angled triangles. There are a few other identities that we use in the case of triangles that are not right-angled.

Sine and Cosine Rule Trigonometric Identities

The sine rule gives the relation between the angles and the corresponding sides of a triangle. For the non-right-angled triangles, we will have to use the sine rule and the cosine rule.

sine rule

The cosine rule gives the relation between the angles and the sides of a triangle and is usually used when two sides and the included angle of a triangle are given.

cosine rule

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Important Notes on Trigonometric Identities

To write the trigonometric ratios of complementary angles, we consider the following as pairs: (sin, cos), (cosec, sec), and (tan, cot).
While writing the trigonometric ratios of supplementary angles, the trigonometric ratio won't change. The sign can be decided using the fact that only sin and cosec are positive in the second quadrant where the angle is of the form (180-θ).
There are 3 formulas for the cos 2x formula. Among them, you can remember just the first one because the other two can be obtained by the Pythagorean identity sin² x + cos²x = 1.
The half-angle formula of tan is obtained by applying the identity tan = sin/cos and then using the half-angle formulas of sin and cos.

Example 1: Can we help Tina prove the following identity using the trigonometric identities?

\(\frac{sin 3 θ+cos 3 θ}{sin θ+cos θ}+sin θ cos θ=1\)

Solution:

We make use of the following identity:

a³+b³= (a+b)(a²-ab+b²)

We use the Pythagorean identities to prove this identity.

\(\begin{align}
&L H S\\[0.2cm] &=\frac{sin 3 θ+cos 3 θ}{sin θ+cos θ}+sin θ cos θ \\[0.2cm]&=\frac{(sin θ+cos θ)\left(sin^ 2 θ-sin θ cos θ+cos^ 2 θ\right)}{sin θ+cos θ}+sin θ cos θ \\[0.2cm]
&=\left(sin ^2 θ-sin θ cos θ+cos ^2 θ\right)+sin θ cos θ \\[0.2cm]
&=sin^2 θ+cos^2 θ\\[0.2cm]&=1\\[0.2cm]&=R H S
\end{align}\)
Example 2:

Can we help Jake prove the following identity using the trigonometric identities?

(sin θ+cosec θ)²+(cos θ+sec θ)²=7+tan² θ+cot² θ

Solution:

We use the reciprocal identities and Pythagorean identities to prove this identity.

L H S

= (sin² θ + cosec² θ + 2 sin θ cosec θ) + (cos² θ + sec² θ + 2 cos θ sec θ)
=sin² θ + cos² θ + cosec²θ + sec²θ + 2 + 2
=1 + (1+cot² θ) + (1+tan² θ) + 2 + 2
=7 + tan² θ + cot² θ = R H S
Example 3:

Can we help James find the exact value of sin75° using the trigonometric identities?

Solution:

We know that 75° = 30°+45°

We apply the sum trigonometric identity of sin to find the value of sin 75°.

sin (A+B) = sin A cos B + cos A sin B

Substitute A=30° and B=45° here on both sides:

sin (30° + 45°) = sin 30° cos 45°+ cos 30° + sin 45°
sin 75° = \(\dfrac{1}{2}\cdot \dfrac{\sqrt{2}}{2}+ \dfrac{\sqrt{3}}{2} \cdot \dfrac{\sqrt{2}}{2}\)
sin 75° = \(\dfrac{\sqrt{2}+ \sqrt{6}}{4}\) =\( \dfrac{\sqrt3 + 1}{2\sqrt2} \)

Here, the values of sin 30°, cos 45°, cos 30°, and sin 45° can be obtained by using the trigonometric table.

Answer: Sin75°= \( \dfrac{\sqrt3 + 1}{2\sqrt2} \)

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FAQs on Trigonometric Identities

What Are the 3 Trigonometric Identities?

The three trigonometric identities are:

sin²θ + cos²θ = 1
1+tan²θ = sec²θ
1+cot²θ = cosec²θ

What Are Trigonometric Identites used for?

The trigonometric identities are used for solving various geometric, trigonometric, and other math problems. They are equations that are always true.

How to Prove Trigonometric Identies?

Trigonometric Identities can be proved by using other known Pythagorean and trigonometric identities. We can use some trigonometric ratios and formulas as well to prove the trig identities.

How Do you Solve equations with Trigonometric Identities?

Various math problems can be solved using trigonometric identities. We can convert equations into Parametric equations and then apply the trigonometric identities to solve them.

What Trigonometric Identities We Must Know?

All trig identities are used in solving the problems. The main trigonometric identities are Pythagorean identities, reciprocal identities, sum and difference identities, double angle and half-angle identities. For the non-right-angled triangles, we will have to use the sine rule and the cosine rule.

What Are Eight Fundamental Trigonometric Identities?

The eight fundamental trigonometric identities are:

cosec θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/ tan θ
sin²θ + cos²θ = 1
tanθ = sinθ/ cos θ
1+ tan²θ = sec²θ
cot θ = cosθ/ sinθ
1+ cot²θ = cosec²θ

What Is the Use of Trigonometric Identities?

We can use trigonometric identities to solve numerous math problems. The identities make the problems easy to solve.

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