The equation $\sin x + \cos x = 2$has

$\sum\limits_{r = 1}^{100} {\frac{{\tan \,{2^{r - 1}}}}{{\cos \,{2^r}}}} $ is equal to

If $1 + \cot \theta = {\rm{cosec}}\theta $, then the general value of $\theta $ is

If $\cot \theta + \tan \theta = 2{\rm{cosec}}\theta $, the general value of $\theta $ is

If $\tan 2\theta \tan \theta = 1$, then the general value of $\theta $ is

Let $f(x)=\cos 5 x+A \cos 4 x+B \cos 3 x$ $+C \cos 2 x+D \cos x+E$, and

$T=f(0)-f\left(\frac{\pi}{5}\right)+f\left(\frac{2 \pi}{5}\right)-f\left(\frac{3 \pi}{5}\right)+\ldots+f\left(\frac{8 \pi}{5}\right)-f\left(\frac{9 \pi}{5}\right) \text {. }$Then, $T$