Question

(Sec^2 54 - cot^2 36) / (cosec^2 57 - tan^2 33) + 2sin^2 38 sec^2 52 - sin^2 45

Answer 1

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Answer 2

The value of [ (sec²54° - cot²36°)/(cosec²57° - tan²33°) + 2sin²38° sec²52° - sin²45°] is 5/2.

We have to find the value of (sec²54° - cot²36°)/(cosec²57° - tan²33°) + 2sin²38° sec²52° - sin²45°

We know, cot(90° - θ) = tanθ

⇒cot36° = cot(90° - 54°) = tan54° ...(1)

Similarly, tan(90° - θ) = cotθ

⇒tan33° = tan(90° - 57°) = cot57° ...(2)

And sec(90° - θ) = cosecθ

∴ sec52° = sec(90° - 38°) = cosec38° ...(3)

Here, (sec²54° - cot²36°)/(cosec²57° - tan²33°) + 2sin²38° sec²52° - sin²45°

from equations (1), (2) and (3),

= (sec²54° - tan²54°)/(cosec²57° - cot²57°) + 2sin²38°cosec²38° - sin²45°

We know,

• sec²x - tan²x = 1 ∴ sec²54° - tan²54° = 1
• cosec²x - cot²x = 1 ∴ cosec²57° - cot²57° = 1
• sinx cosec x = 1 ∴ sin²38° cosec²38° = 1
• sin45° = 1/√2

= 1/1 + 2(1) - (1/√2)²

= 1 + 2 - 1/2

= 5/2

Therefore the value of given trigonometric expression is 5/2.