Trigonometric Equations - Trigonometry - Mathematics Previous Year Questions

The number of distinct solutions of the equation \({5 \over 4}{\cos ^2}2x + {\cos ^4}x + {\sin ^4}x + {\cos ^6}x = 2\) in the interval [0, 2\(\pi\)] is

The equation \((\cos \beta - 1){x^2} + (\cos \beta )x + \sin \beta = 0\) in the variable x has real roots, then \(\beta\) lies in the interval

Total number of solutions of \(\left| {\cot x} \right| = \cot x + {1 \over {\sin x}},x \in [0,3\pi ]\) is equal to

If \({\cos ^3}x\,.\,\sin 2x = \sum\limits_{m = 1}^n {{a_m}\sin mx}\) is identity in x, then

The sum of all the solution of the equation \(\cos \theta \cos \left( {{\pi \over 3} + \theta } \right)\cos \left( {{\pi \over 3} - \theta } \right) = {1 \over 4},\theta \in [0,6\pi ]\)

If \(n\) is the number of solutions of the equation \(2 \cos x\left(4 \sin \left(\frac{\pi}{4}+x\right) \sin \left(\frac{\pi}{4}-x\right)-1\right)=1, x \in[0, \pi]\) and \(S\) is the sum of all these solutions, then the ordered pair \((n, S)\) is