(tan θ + 2) (2 tan θ + 1) = 5 tan θ + sec2 θ. Write ‘True’ or ‘False’ and justify your answer
Solution:
Given, the expression is (tan θ + 2) (2 tan θ + 1) = 5 tan θ + sec² θ.
We have to determine if the given expression is true or not.
LHS: (tan θ + 2) (2 tan θ + 1)
By multiplicative and distributive property,
= 2tan²θ + tanθ + 4tanθ + 2
= 2tan²θ + 5tanθ + 2
By using trigonometric ratio of complementary angles,
1 + tan² A = sec² A
tan² A = sec² A - 1
So, 2(sec²θ - 1) + 5tanθ + 2
= 2sec²θ - 2 + 5tanθ + 2
= 2sec²θ + 5tanθ
RHS = 5 tan θ + sec² θ
LHS ≠ RHS
✦ Try This: Prove that 1/(secθ - tanθ) = secθ + tanθ
We have to prove that 1/(secθ - tanθ) = secθ + tanθ.
LHS = 1/(secθ - tanθ)
Multiplying and dividing by secθ + tanθ,
= 1/(secθ - tanθ) × (secθ + tanθ)/(secθ + tanθ)
= (secθ + tanθ)/(sec²θ - tan²θ)
By using trigonometric identity,
sec² A - tan² A = 1
So, (secθ + tanθ)/(sec²θ - tan²θ)
= (secθ + tanθ)/1
= secθ + tanθ
= RHS
LHS = RHS
Therefore, 1/(secθ - tanθ) = secθ + tanθ
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 8
NCERT Exemplar Class 10 Maths Exercise 8.2 Problem 6
(tan θ + 2) (2 tan θ + 1) = 5 tan θ + sec2 θ. Write ‘True’ or ‘False’ and justify your answer
Summary:
Trigonometric identities are the equalities involving trigonometric functions and hold true for every value of the variables involved, such that both sides of the equality are defined. The statement “(tan θ + 2) (2 tan θ + 1) = 5 tan θ + sec² θ” is false
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