Question

Sec^254°-cot^236°+2sin^238°sec^252°-sin^245°/cosec^257°-tan^233°

Answer 1

$\frac{sec^{2} 54\degree - cot^{2} 36\degree+ 2sin^{2} 38\degree sec^{2}52\degree - sin^{2} 45\degree }{cosec^{2} 57\degree - tan^{2} 33\degree }$

$= \frac{sec^{2} (90 - 36\degree) - cot^{2} 36\degree+ 2sin^{2} 38\degree sec^{2}(90- 38\degree) - sin^{2} 45\degree }{cosec^{2} (90- 33\degree) - tan^{2} 33\degree }$

$= \frac{(cosec^{2} 36\degree- cot^{2} 36\degree)+ 2sin^{2} 38\degree cosec^{2} 38\degree - sin^{2} 45\degree }{sec^{2} 33\degree - tan^{2} 33\degree }$

$= \frac{ 1 + 2 sin^{2} 38\degree \times \frac{1}{sin^{2} 38\degree } - \Big( \frac{1}{\sqrt{2}}\Big)^{2} } { 1 }$

" ____________________

By Trigonometric Identities :

$\pink { i ) cosec^{2} \theta - cot^{2} \theta = 1 }$

$\blue { ii ) sec^{2} \theta - tan^{2} \theta = 1 }$

_______________________"

$= 1 + 2 - \frac{1}{2}$

$= 3 - \frac{1}{2} \\= \frac{6-1}{2} \\= \frac{5}{2}$

Therefore.,

$\red{\frac{sec^{2} 54\degree - cot^{2} 36\degree+ 2sin^{2} 38\degree sec^{2}52\degree - sin^{2} 45\degree }{cosec^{2} 57\degree - tan^{2} 33\degree }}$

$\green {= \frac{5}{2}}$

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