Question

Prove the identity, $\dfrac{\cos {{9}^{\circ }}+\sin {{9}^{\circ }}}{\cos {{9}^{\circ }}-\sin {{9}^{\circ }}}=\cot {{36}^{\circ }}$

Hint: Divide numerator and denominator by $\cos {{9}^{\circ }}$ or $\sin {{9}^{\circ }}$ to get a trigonometric identity.

Use relation $\cot \left( 90-\theta \right)=\tan \theta$ or $\tan \left( 90-\theta \right)=\cot \theta$ for simplifying it further.

We have to prove that,
$\dfrac{\cos {{9}^{\circ }}+\sin {{9}^{\circ }}}{\cos {{9}^{\circ }}-\sin {{9}^{\circ }}}=\cot {{36}^{\circ }}..........\left( i \right)$
Let us prove the given relation by simplifying LHS of equation (i) we have,
LHS$=\dfrac{\cos {{9}^{\circ }}+\sin {{9}^{\circ }}}{\cos {{9}^{\circ }}-\sin {{9}^{\circ }}}$
Dividing numerator and denominator by $\cos {{9}^{\circ }}$, we get,
LHS = $\dfrac{\dfrac{\cos {{9}^{\circ }}+\sin {{9}^{\circ }}}{\cos {{9}^{\circ }}}}{\dfrac{\cos {{9}^{\circ }}-\sin {{9}^{\circ }}}{\cos {{9}^{\circ }}}}$
Now, we can divide $\cos {{9}^{\circ }}$ by $\cos {{9}^{\circ }}$ and $\sin {{9}^{\circ }}$ by $\cos {{9}^{\circ }}$ in numerator and similarly, $\cos {{9}^{\circ }}$ by $\cos {{9}^{\circ }}$ and $\sin {{9}^{\circ }}$ by $\cos {{9}^{\circ }}$ in denominator as well. We get,
LHS = $\dfrac{\dfrac{\cos {{9}^{\circ }}}{\cos {{9}^{\circ }}}+\dfrac{\sin {{9}^{\circ }}}{\cos {{9}^{\circ }}}}{\dfrac{\cos {{9}^{\circ }}}{\cos {{9}^{\circ }}}-\dfrac{\sin {{9}^{\circ }}}{\cos {{9}^{\circ }}}}............(ii)$
We have trigonometric identity,
$\tan \theta =\dfrac{\sin \theta }{\cos \theta }...........(iii)$
Hence, equation (ii) can be simplified as
LHS = $\dfrac{1+\tan {{9}^{\circ }}}{1-\tan {{9}^{\circ }}}=\dfrac{1+\tan {{9}^{\circ }}}{1-\tan {{9}^{\circ }}\left( 1 \right)}$
As we know, $\tan {{45}^{\circ }}=1$ , so above equation can be written as
LHS = $\dfrac{\tan {{45}^{\circ }}+\tan {{9}^{\circ }}}{1-\tan {{9}^{\circ }}\tan {{45}^{\circ }}}.............\left( iv \right)$
Now, using trigonometric identity,
$\tan \left( A+B \right)=\dfrac{\tan A+\tan B}{1-\tan A+\tan B}$
Hence equation (iv) can be rewritten with the help of the above equation. So, we get
LHS = $\tan \left( 45+9 \right)=\tan {{54}^{\circ }}$
Now, we can use relation $\tan \left( {{90}^{\circ }}-\theta \right)=\cot \theta$ to convert the ‘tan’ function to ‘cot’.
Hence, LHS can be given as
LHS = $\tan \left( 90-36 \right)=\cot {{36}^{\circ }}$
Therefore, LHS = RHS = $\cot {{36}^{\circ }}$
Hence proved the given relation

Note: Another approach for the given question would be to use trigonometric identities
$\cos C+\cos D=2\cos \dfrac{C+D}{2}\cos \dfrac{C-D}{2}$
$\cos C-\cos D=-2\sin \dfrac{C+D}{2}\sin \dfrac{C-D}{2}$
Use relation $\cos \theta =\sin \left( 90-\theta \right)$
So, LHS = $\dfrac{\cos {{9}^{\circ }}+\cos {{81}^{\circ }}}{\cos 9-\cos 81}$
One can go wrong while converting $\dfrac{\tan 45+\tan {{9}^{\circ }}}{1-\tan 9\tan 45}$ to $\tan {{54}^{\circ }}$ .
He or she may confuse between formula tan(A-B) and tan(A+B).