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