QUESTION

# When expressed in terms of angles between ${{0}^{\circ }}$ to ${{30}^{\circ }}$, then, $\cos {{75}^{\circ }}+\cot {{75}^{\circ }}$, becomes(a) $\sin {{15}^{\circ }}+\tan {{15}^{\circ }}$ (b) $\sin {{15}^{\circ }}+\cos {{15}^{\circ }}$(c) $\cos {{15}^{\circ }}+\tan {{15}^{\circ }}$ (d) None of these

Hint:To rewrite the given expression in terms of angles between ${{0}^{\circ }}$ to ${{30}^{\circ }}$, use the trigonometric property of cosine and cotangent functions which are $\cos x=\sin \left( {{90}^{\circ }}-x \right)$ and$\cot x=\tan \left( {{90}^{\circ }}-x \right)$. Simplify the expression and choose the correct option.

We have to rewrite the expression $\cos {{75}^{\circ }}+\cot {{75}^{\circ }}$ in terms of angles between ${{0}^{\circ }}$ to ${{30}^{\circ }}$.
We know that $\cos x=\sin \left( {{90}^{\circ }}-x \right)$.
Substituting $x={{75}^{\circ }}$ in the above equation, we have $\cos {{75}^{\circ }}=\sin \left( {{90}^{\circ }}-{{75}^{\circ }} \right)$.
Thus, we have $\cos {{75}^{\circ }}=\sin \left( {{15}^{\circ }} \right).....\left( 1 \right)$.
We also know that $\cot x=\tan \left( {{90}^{\circ }}-x \right)$.
Substituting $x={{75}^{\circ }}$ in the above equation, we have $\cot {{75}^{\circ }}=\tan \left( {{90}^{\circ }}-{{75}^{\circ }} \right)$.
Thus, we have $\cot {{75}^{\circ }}=\tan \left( {{15}^{\circ }} \right).....\left( 2 \right)$.
Substituting equation (1) and (2) in the given expression, we can rewrite $\cos {{75}^{\circ }}+\cot {{75}^{\circ }}$ as $\cos {{75}^{\circ }}+\cot {{75}^{\circ }}=\sin {{15}^{\circ }}+\tan {{15}^{\circ }}$.
Hence, we can rewrite the expression $\cos {{75}^{\circ }}+\cot {{75}^{\circ }}$ as $\sin {{15}^{\circ }}+\tan {{15}^{\circ }}$, which is option (a).