Embibe Experts Solutions for Exercise 6: Assignment

If$x,y,z\in \left(-1,1\right)$ satisfy ${\cos }^{-1}x+{\cos }^{-1}y+{\cos }^{-1}z=\pi$, then $x,y,z$ satisfy

If $x^3-x^2+{\tan }^2\alpha x-{\tan }^2\alpha =0$, has positive roots $x_1,x_2,x_3$ then which of following are true?

Solution(s) of the equation $\sin \left(-\frac{\pi }{3}+{\tan }^{-1}x+{\cot }^{-1}x\right)=\frac{1}{2}$ is/are

Let $\left(a,b\right)$ satisfy following equation ${\tan }^{-1}a+{\tan }^{-1}b=\frac{\pi }{4}$

${\tan }^{-1}\frac{1}{a}+{\tan }^{-1}b={\tan }^{-1}\left(7\right)$ then

If ${\tan }^{-1}x+{\tan }^{-1}y-{\sin }^{-1}x=\theta$, then which of the following are true?

The value(s) of $\theta$ satisfying the equation $\theta ={\tan }^{-1}\left(2{\tan }^2\theta \right)-\frac{1}{2}\left({\sin }^{-1}\left(\frac{3\sin 2\theta }{5+4\cos 2\theta }\right)\right)$ is

$2{\tan }^{-1}\left\{\tan \frac{\alpha }{2}\tan \left(\frac{\pi }{4}-\frac{\beta }{2}\right)\right\}$, $\alpha ,\beta \in \left(0,\frac{\pi }{2}\right)$ is equal to

An ordered triplet solution $(x, y, z)$ with $x,y,z\in (0,1)$ and satisfying $x^2+y^2+z^2+2xyz=1$ is -