The value of $x$ that satisfies the relation $x = 1 - x + x^2 - x^3 + x^4 - x^5 + ......... \infty$

If $\sin A = n\sin B,$ then $\frac{{n - 1}}{{n + 1}}\tan \,\frac{{A + B}}{2} = $

If $\sin x + \cos x = \frac{1}{5},$ then $\tan 2x$ is

If $\alpha $ and $\beta $ are solutions of $sin^2\,x + a\, sin\, x + b = 0$ as well that of $cos^2\,x + c\, cos\, x + d = 0$ , then $sin\,(\alpha + \beta )$ is equal to

If $\tan x = \frac{{2b}}{{a - c}}(a \ne c),$

$y = a\,{\cos ^2}x + 2b\,\sin x\cos x + c\,{\sin ^2}x$

and $z = a{\sin ^2}x - 2b\sin x\cos x + c{\cos ^2}x,$ then

$\frac{{\tan A + \sec A - 1}}{{\tan A - \sec A + 1}} = $