If-sin-x-cos-x-tan-x-cosec-x-cot-x-sec-x-7-then-find-the-value-of-sin-2x-

Question Number 19760 by Tinkutara last updated on 15/Aug/17

If \sin x + \cos x + \tan x + cosec x +

\cot x + \sec x = 7, then find the value of

\sin 2 x .

Answered by ajfour last updated on 15/Aug/17

\sin x + \cos x + \frac{1}{\sin x} + \frac{1}{\cos x} + \frac{\sin x}{\cos x}

+ \frac{\cos x}{\sin x} = 7

\Rightarrow (\sin x + \cos x) [1 + \frac{2}{\sin 2 x}] + \frac{2}{\sin 2 x} = 7

let \sin 2 x = t

and as {(\sin x + \cos x)}^{2} = 1 + \sin 2 x

\Rightarrow (\sin x + \cos x) = \sqrt{1 + t}

\Rightarrow \sqrt{1 + t} (1 + \frac{2}{t}) + \frac{2}{t} = 7

\Rightarrow \sqrt{1 + t} (1 + \frac{2}{t}) + (1 + \frac{2}{t}) = 8

\Rightarrow (1 + \frac{2}{t}) (\sqrt{1 + t} + 1) = 8

\Rightarrow (\frac{t + 2}{t}) (\frac{t}{\sqrt{1 + t} - 1)} = 8

\Rightarrow (t + 2) = 8 \sqrt{1 + t} - 8

\Rightarrow 64 (1 + t) = {(t + 10)}^{2}

\Rightarrow t^{2} - 44 t + 36 = 0

\Rightarrow {(t - 22)}^{2} = 484 - 36

\Rightarrow t - 22 = \pm \sqrt{{(21)}^{2} + 7}

\Rightarrow So \sin 2 x = t = 22 - \sqrt{{(21)}^{2} + 7} .

Commented by Tinkutara last updated on 15/Aug/17

Thank you very much Sir!