Question

Please answer this step by step.

Answer 1

Step-by-step explanation:

LHS =

$\frac{ {(1 + \sin(x)) }^{2} +{( 1 - \sin(x)) }^{2} }{2 \cos {}^{2} (x) }$

$we \: have \:$

${(a + b)}^{2} + {(a - b)}^{2} =2({a}^{2} + {b}^{2} )$

Using this,the expression becomes

$\frac{2( {1}^{2} + \sin {}^{2} (x) )}{2 \cos {}^{2} (x) }$

$= \frac{1 + \sin {}^{2} (x) }{ \cos {}^{2} (x) }$

We know that

$\sin {}^{2} (x) + \cos {}^{2} (x) = 1$

$or \: \cos {}^{2} (x) = 1 - \sin {}^{2} (x)$

Using this, the expression becomes

$\frac{1 + \sin {}^{2} (x) }{1 - \sin {}^{2} (x) }$

= RHS

Hence proved