Question

Please help me solve this question on trigonometry ​

Answer 1

Step-by-step explanation:

Let A= α,

Then,

$x = \cot( \alpha ) + \cos( \alpha ) \: \: and \: \: y = \cot( \alpha ) - \cos( \alpha )$

Now,

${ (\frac{x - y}{x + y}) }^{2} + { (\frac{x - y}{2} )}^{2}$

$= (\frac{ \cot( \alpha ) + \cos( \alpha ) - \cot( \alpha ) + \cos( \alpha ) }{ \cot( \alpha ) + \cos( \alpha ) + \cot( \alpha ) - \cos( \alpha ) } )^{2} + { (\frac{ \cot( \alpha ) + \cos( \alpha ) - \cot( \alpha ) + \cos( \alpha ) }{2} })^{2}$

$= ({ \frac{2 \cos( \alpha ) }{2 \cot( \alpha ) } })^{2} + ({ \frac{2 \cos( \alpha ) }{2} })^{2}$

$= \frac{ \cos^{2} ( \alpha ) }{ \cot ^{2} ( \alpha ) } + \cos^{2} ( \alpha )$

$= \sin^{2} ( \alpha ) + \cos^{2} ( \alpha )$

$= 1$

Hence proved