The equation $2{\cos ^2}\left( {\frac{x}{2}} \right)\,{\sin ^2}x\, = \,{x^2}\, + \,\frac{1}{{{x^2}}},\,0\,\, \leqslant \,\,x\,\, \leqslant \,\,\frac{\pi }{2}\,\,$ has

The general solution of $sin\, x + sin \,5x = sin\, 2x + sin \,4x$ is :

The number of elements in the set $S=\left\{x \in R : 2 \cos \left(\frac{x^{2}+x}{6}\right)=4^{x}+4^{-x}\right\}$ is$.....$

One root of the equation $\cos x - x + \frac{1}{2} = 0$ lies in the interval

If $\sin 2\theta = \cos 3\theta $ and $\theta $ is an acute angle, then $\sin \theta $ is equal to

The value of expression $\frac{{2(\sin {1^o} + \sin {2^o} + \sin {3^o} + ..... + \sin {{89}^o})}}{{2(\cos {1^o} + \cos {2^o} + .... + \cos {{44}^o}) + 1}}$ equals