Question

Prove that
1/tanA+cotA=cosAsinA

Answer 1

To Prove:

• $\frac{1}{ \tan(</strong><strong>A</strong><strong>) + \cot(</strong><strong>A</strong><strong>) } = \cos(</strong><strong>A</strong><strong>) \sin( </strong><strong>A</strong><strong> )$

Proof:

We have ,

$L.H.S = \frac{1}{ \tan( A ) + \cot( A ) }$

But,

we know that,

$\tan( A) = \frac{ \sin( A ) }{ \cos( A ) } \\ \\ and \\ \\ \cot( A ) = \frac{ \cos( A ) }{ \sin( A ) }$

Therefore,

pUtting the values,

we get,

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

But,

we know that,

${ \sin }^{2} \alpha + { \cos }^{2} \alpha = 1$

Therefore,

putting the values,

we get,

$= \frac{ \sin( A) \cos( A ) }{1} \\ \\ = \sin( A ) \cos( A ) =R.H.S$

Thus,

L.H.S = R.H.S

Hence, Proved