Solveeit Logo
Login

Question

Mathematics Question on Properties of Inverse Trigonometric Functions

The value of 2cot⁡−112−cot⁡−1432 \cot ^{-1} \frac{1}{2}-\cot ^{-1} \frac{4}{3} is

A

−π8-\frac{\pi}{8}

B

3π2\frac{3\pi}{2}

C

π4\frac{\pi}{4}

D

π2\frac{\pi}{2}

Answer

π2\frac{\pi}{2}

Explanation

Solution

2cot⁡−112−cot⁡−1432 \cot ^{-1} \frac{1}{2}-\cot ^{-1} \frac{4}{3}
=2tan⁡−12−tan⁡−134=2 \tan ^{-1} 2-\tan ^{-1} \frac{3}{4}
=π+tan⁡−14−3−tan⁡−134=\pi+\tan ^{-1} \frac{4}{-3}-\tan ^{-1} \frac{3}{4}
=π−tan⁡−143−tan⁡−134=\pi-\tan ^{-1} \frac{4}{3}-\tan ^{-1} \frac{3}{4}
=\pi-\left\\{\tan ^{-1} \frac{\frac{4}{3}+\frac{3}{4}}{1-\frac{4}{3} \cdot \frac{3}{4}}\right\\}
=π−tan⁡−1∞=\pi-\tan ^{-1} \infty
=π−π2=π2=\pi-\frac{\pi}{2}=\frac{\pi}{2}