Trigonometric Equations - Trigonometry - Mathematics Previous Year Questions

$1+\cos x+\cos ^2 x+\cos ^3 x+\ldots$ to $\infty=4+2 \sqrt{3}$, then $x=$

The general solution of the equation $\sqrt{6-5 \cos x+7 \sin ^2 x}-\cos x=0$ also satisfies the equation

If $2 \sin \theta+3 \cos \theta=2$ and $\theta \neq(2 n+1) \frac{\pi}{2}$, then $\sin \theta+\cos \theta=$

If $x \in(-\pi, \pi)$, then the number of solutions of the equation $2 \sin x \sin 3 x \sin 5 x+\sin 5 x \cos 4 x=0$ is

If $\cos \alpha+\cos \beta+\cos \gamma=0=\sin \alpha+\sin \beta+\sin \gamma$, then $\sin 2 \alpha+\sin 2 \beta+\sin 2 \gamma=$

Number of solutions of the equation $\sin ^2 \theta+2 \cos ^2 \theta-\sqrt{3} \sin \theta \cos \theta=2$ lying in the interval ( $-\pi, \pi$ ) is

If $a, b$ are real numbers and $\alpha$ is a real roots of $x^2+12+3 \sin (a+b x)+6 x=0$, then the value of $\cos (a+b \alpha)$ for the least positive value of $a+b \alpha$ is

Number of solutions of the equation $\tan ^2 x+3 \cot ^2 x=2 \sec ^2 x$ lying in the interval $[0,2 \pi]$ is

If $0 \leq A, B \leq \frac{\pi}{4}$ and $\cot A+\cot B+\tan A+ \tan B=\cot A \cot B-\tan A \tan B$, then $\sin (A+B)=$