The number of triangles in which $\tan A + \tan B + \tan C = \cot A + \cot B + \cot C$ is:

In a $\triangle ABC$,$(b^2-c^2) \cot A + (c^2-a^2) \cot B =$

Let $S = \{\theta \in (0, 2\pi) : 7 \cos^2 \theta - 3 \sin^2 \theta - 2 \cos^2 2\theta = 2\}$. Then,the sum of roots of all the equations $x^2 - 2(\tan^2 \theta + \cot^2 \theta)x + 6 \sin^2 \theta = 0$ for $\theta \in S$ is...

The equation $\sin^4 x + \cos^4 x + \sin 2x + \alpha = 0$ is solvable for

For a triangle $ABC$,the value of $\cos 2A + \cos 2B + \cos 2C$ is least. If its inradius is $3$ and incentre is $M$,then which of the following is $NOT$ correct?

If $\alpha$ and $\beta$ are different values of $x$ satisfying $a \cos x + b \sin x = c,$ then $\tan \left( \frac{\alpha + \beta}{2} \right) = $