Question

Prove that ( 1 + sin A / cos A) + cos A /( 1+ sin A) = 2 cos A ​

Answer 1

Correct question

$\rm \implies \dfrac{1 + \sin A}{ \cos A} + \dfrac{ \cos A}{1 + \sin A} = 2 \sec A$

Solution:-

$\rm \implies \dfrac{1 + \sin A}{ \cos A} + \dfrac{ \cos A}{1 + \sin A} = 2 \sec A$

Taking lcm

$\rm\implies \dfrac{(1 + \sin A)(1 + \sin A) + \cos A \times \cos A}{ \cos A(1 + \sin A)}$

$\rm \implies \dfrac{(1 + \sin A) ^{2} + \cos {}^{2} A }{ \cos A + \sin A \times \cos A}$

Using this identities (a² + b² ) = a² + b² + 2ab

$\rm \implies \dfrac{1 + \sin {}^{2} A + 2 \sin A + \cos {}^{2} A}{\cos A(1 + \sin A) }$

$\rm \implies \dfrac{1 + 1 + 2 \sin A }{\cos A(1 + \sin A) }$

$\rm \implies \dfrac{2 + 2 \sin A }{\cos A(1 + \sin A) }$

$\rm \implies \dfrac{2(1 + \sin A ) }{\cos A(1 + \sin A) }$

$\rm \implies \dfrac{2 \cancel{(1 + \sin A )} }{\cos A \cancel{(1 + \sin A) }}$

$\rm \implies \dfrac{2 }{\cos A}$

$\rm \: 2 \sec A$