What is the value of 1 + 2 ^2 ( 90^ - x ) - 2 cosec ( 90^ - x ) ( 90^ - x ) cosec ( 90^ - x ) - ( 90^ - x )?

Option D. secx - tanx. Answer & Solution Answer: Option D Solution: & 1 + 2 ^2 ( 90^ - x ) - 2 cosec ( 90^ - x ) ( 90^ - x ) cosec ( 90^ - x ) - ( 90^ - x ) & 1 + 2 ^2x - 2 x x x - x & 1 + 2 ( ^2x - 1 ) - 2 x x x - x & 2 ^2x - 1 - 2 x x x - x & 2 x ( x - x ) x - x - 1 x - x & 2 x - 1 x - x & 2 x - 1 x…

What is the value of 1 + 2 ^2 ( 90^ - x ) - 2 cosec ( 90^ - x ) ( 90^…

Answer: Option D

Solution:

\frac{1 + 2 cot ^{2} ( 90 ^{\circ} - x ) - 2 cosec ( 90 ^{\circ} - x ) cot ( 90 ^{\circ} - x )}{cosec ( 90 ^{\circ} - x ) - cot ( 90 ^{\circ} - x )} \Rightarrow \frac{1 + 2 tan ^{2} x - 2 sec x tan x}{sec x - tan x} \Rightarrow \frac{1 + 2 ( sec ^{2} x - 1 ) - 2 sec x tan x}{sec x - tan x} \Rightarrow \frac{2 sec ^{2} x - 1 - 2 sec x tan x}{sec x - tan x} \Rightarrow \frac{2 sec x ( sec x - tan x )}{sec x - tan x} - \frac{1}{sec x - tan x} \Rightarrow 2 sec x - \frac{1}{sec x - tan x} \Rightarrow 2 sec x - \frac{1}{sec x - tan x} \times \frac{sec x + tan x}{sec x + tan x} \Rightarrow 2 sec x - \frac{sec x + tan x}{sec ^{2} x - tan ^{2} x} \Rightarrow 2 sec x - sec x - tan x \Rightarrow sec x - tan x Alternative : \frac{1 + 2 cot ^{2} ( 90 ^{\circ} - x ) - 2 cosec ( 90 ^{\circ} - x ) cot ( 90 ^{\circ} - x )}{cosec ( 90 ^{\circ} - x ) - cot ( 90 ^{\circ} - x )} By putting x = 4 5^{\circ} in equation \Rightarrow \frac{1 + 2 tan ^{2} 45 ^{\circ} - 2 sec 45 ^{\circ} tan 45 ^{\circ}}{sec 45 ^{\circ} - tan 45 ^{\circ}} \Rightarrow \frac{1 + 2 - 2 2}{2 - 1} \Rightarrow \frac{3 - 2 2}{2 - 1} \times \frac{( 2 + 1 )}{( 2 + 1 )} \Rightarrow 32 - 22 - 1 \Rightarrow 2 By satisfying in options \Rightarrow sec x - tan x \Rightarrow 2 - 1