Question

if sin theta + cos theta =root 3, then prove that tan theta + cot theta =1​

Answer 1

Step-by-step explanation:

Given

$\sin( \alpha ) + \cos( \alpha ) = \sqrt{3}$

Squarring and adding, we get

$= > {( \sin \alpha + \cos \alpha ) }^{2} = {( \sqrt{3})} ^{2} \\ \\ = > { \sin }^{2} \alpha + { \cos}^{2} \alpha + 2 \sin( \alpha ) \cos( \alpha ) = 3 \\ \\ = > 1 + 2 \sin( \alpha ) \cos( \alpha ) = 3 \\ \\ = > 2 \sin( \alpha ) \cos( \alpha ) = 3 - 1 \\ \\ = > \sin( \alpha ) \cos( \alpha ) = \frac{2}{2} \\ \\ = > \sin( \alpha ) \cos( \alpha ) = 1$

Now, to find the value of

$\tan( \alpha ) + \cot( \alpha )$

Solving further, we get

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

Hence, Proved