2023 · Q46
Consider the following :
- 1cos⁴θ -sin⁴θ =(2tanθ )/(1-tan²θ ),0
- 2cosecθ +cotθ =(1)/(cosecθ -cotθ ),0
- 3cos²θ -sin²θ =(1-tan²θ )/(1+tan²θ ),0
Which of the above equations are identities?
Elementary MathematicsTrigonometry
- A1 and 2 only
- B2 and 3 only
- C1 and 3 only
- D1, 2 and 3
Explanation
(1) cos⁴θ − sin⁴θ = cos2θ while the RHS equals tan2θ — not an identity. (2) (cosecθ − cotθ)(cosecθ + cotθ) = cosec²θ − cot²θ = 1, so cosecθ + cotθ = 1/(cosecθ − cotθ) is an identity. (3) cos²θ − sin²θ = cos2θ = (1 − tan²θ)/(1 + tan²θ) is an identity. Only 2 and 3 are identities (B).