Question & Answer
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

ANSWER Verified Verified
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.

Complete step-by-step answer:
We have to rewrite the expression $\cos {{75}^{\circ }}+\cot {{75}^{\circ }}$ in terms of angles between ${{0}^{\circ }}$ to ${{30}^{\circ }}$.
We will use trigonometric properties of cosine and cotangent functions to simplify the given expression.
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).

Note: Trigonometric functions are real functions that relate any angle of a right-angled triangle to the ratios of any two of its sides. The widely used trigonometric functions are sine, cosine, and tangent. However, we can also use their reciprocals, i.e., cosecant, secant, and cotangent. We can use geometric definitions to express the value of these functions on various angles using unit circle (circle with radius 1). We also write these trigonometric functions as infinite series or as solutions to differential equations. Thus, allowing us to expand the domain of these functions from the real line to the complex plane. One should be careful while using the trigonometric identities and rearranging the terms to convert from one trigonometric function to the other one.