2021 · Q53
Consider the following :
- 1sin⁴θ − sin²θ = cos⁴θ − cos²θ.
- 2sin⁴θ + cos⁴θ = 1 + 2sin²θcos²θ.
- 3tan⁴θ + tan²θ = sec⁴θ − sec²θ.
Which of the above are identities ?
Elementary MathematicsTrigonometry
- A1 and 2 only
- B2 and 3 only
- C1 and 3 only
- D1, 2 and 3
Explanation
Statement 1: both sides equal sin²θ(sin²θ − 1) = cos²θ(cos²θ − 1) = −sin²θcos²θ, so it is an identity. Statement 2 is false: sin⁴θ + cos⁴θ = 1 − 2sin²θcos²θ, not 1 + 2sin²θcos²θ. Statement 3: both sides equal tan²θ(tan²θ + 1) = sec²θ(sec²θ − 1) = tan²θsec²θ, so it is an identity. Hence 1 and 3 only, option (c).