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If cos²α – sin²α = tan²β, then the value of cos²β – sin²β is
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- cot²α
- cot²α
- tan²α
- tan²α
Correct Option: C
cos²α – sin²α = tan²β
⇒ cos²α – (1 – cos²α) = tan²β
⇒ 2cos²α – 1 = tan²β
⇒ 2cos²α = 1 + tan²β = sec²β
| ⇒ cos²β = | 2cos²α |
sin²β = 1 – cos²β
| = 1 - | 2cos²α |
| = | 2cos²α |
∴ cos²β – sin²β
| = | - | 2cos²α | 2cos²α |
| = | 2cos²α |
| = | = | 2cos²α | cos²α |
= tan²α
Note : It is an identity.
