Question

Value of $\cos (-{1230}^{\circ })\times \cot (-{315}^{\circ })$ is:

• (A) $\frac{-\sqrt{3}}{2}$
• (B) $\frac{2}{\sqrt{3}}$
• (C) $\frac{1}{2}$
• (D) $-\frac{1}{2}$

Answer 1

The Correct Option is A

Explanation:
Given:$\cos (-{1230}^{\circ })\times \cot (-{315}^{\circ })$Concept:$\cos (-\theta )=\cos \theta$$\cos (2n\pi +\theta )=\cos (\theta )$$\cos (\pi -\theta )=-\cos (\theta )$$\cot (-\theta )=-\cot (\theta )$$\cot (2n\pi -\theta )=-\cot (\theta )$According to the ques,$\cos (-{1230}^{\circ })=\cos \left({1230}^{\circ }\right)$$(\because \cos (-\theta )=\cos \theta )$$=\cos (3\times {360}^{\circ }+150)$$(\because \cot (2n\pi -\theta )=-\cot (\theta ))$$=\cos \left({150}^{\circ }\right)$$=\cos ({180}^{\circ }-{30}^{\circ })$$=-\cos {30}^{\circ }(\because \cos (\pi -\theta )=-\cos (\theta ))$$=\frac{-\sqrt{3}}{2}$Now,$\cot (-{315}^{\circ })=-\cot \left({315}^{\circ }\right)(\because \cot (-\theta )=-\cot (\theta ))$$=-\cot ({360}^{\circ }-{45}^{\circ })$$=\cot {45}^{\circ }(\because \cot (2n\pi -\theta )=-\cot (\theta ))$$=1$$\cos (-{1230}^{\circ })\times \cot (-{315}^{\circ })=\frac{-\sqrt{3}}{2}$Hence, the correct option is (A).