The value of $\operatorname{cosec}10^\circ-\sqrt3\sec10^\circ$ is equal to:

Question

Accepted Answer

C. 1. Start with the given expression:
   $$
   \operatorname{cosec}10^\circ-\sqrt3\sec10^\circ
   =\frac{1}{\sin10^\circ}-\frac{\sqrt3}{\cos10^\circ}.
   $$

2. Take the LCM:
   $$
   =\frac{\cos10^\circ-\sqrt3\sin10^\circ}
   {\sin10^\circ\cos10^\circ}.
   $$

3. Multiply the numerator and denominator appropriately:
   $$
   =4\left[
   \frac{\frac12\cos10^\circ-\frac{\sqrt3}{2}\sin10^\circ}
   {2\sin10^\circ\cos10^\circ}
   \right].
   $$

4. Use the identities
   $$
   \sin(A-B)=\sin A\cos B-\cos A\sin B,
   $$
   and
   $$
   2\sin10^\circ\cos10^\circ=\sin20^\circ.
   $$

Since
   $$
   \frac12=\sin30^\circ,\qquad
   \frac{\sqrt3}{2}=\cos30^\circ,
   $$
   the numerator becomes
   $$
   \sin(30^\circ-10^\circ)=\sin20^\circ.
   $$

5. Therefore,
   $$
   4\left[
   \frac{\sin20^\circ}{\sin20^\circ}
   \right]
   =4.
   $$

6. Hence, the required value is
   $$
   \boxed{4}.
   $$

Therefore, **Option C** is correct.

The value of cosec⁡10∘−3sec⁡10∘\operatorname{cosec}10^\circ-\sqrt3\sec10^\circcosec10∘−3​sec10∘ is equal to:

The value of $\operatorname{cosec}10^\circ-\sqrt3\sec10^\circ$ is equal to: