If sin x + cos x + tan x + cot x + sec x + csc x = 7 , find sin2x?


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Question Number 209838 by malwan last updated on 23/Jul/24

$I f s i n x + c o s x + t a n x + c o t x +$ $s e c x + c s c x = 7, f i n d s i n 2 x ?$
	Answered by mahdipoor last updated on 23/Jul/24

	$= \frac{s^{2} c + c^{2} s + s^{2} + c^{2} + c + s}{s c} = \frac{1 + (s c + 1) (c + s)}{s c} =$ $\frac{1 + (\frac{s i n (2 x)}{2} + 1) (c o s x + s i n x)}{\frac{s i n (2 x)}{2}} = 7 (e q i)$ $g e t s i n (2 x) = 2 u$ $\Rightarrow s i n x + c o s x = \sqrt{{s i n}^{2} x + {c o s}^{2} x + 2 s i n (x) c o s (x)} =$ $\sqrt{1 + s i n (2 x)} = \sqrt{1 + 2 u}$ $e q i \Rightarrow \frac{1 + (u + 1) \sqrt{1 + 2 u}}{u} = 7 \overset{u \neq 0}{\Rightarrow} \sqrt{1 + 2 u} = \frac{7 u - 1}{u + 1} \Rightarrow$ $1 + 2 u = \frac{49 u^{2} + 1 - 14 u}{u^{2} + 1 + 2 u} \Rightarrow 2 u^{3} - 44 u^{2} + 18 u = 0$ $u = 0 (r e j e c t), 11 \pm 4 \sqrt{7} \Rightarrow - 1 ⩽ s i n (2 x) = 2 u ⩽ 1$ $\Rightarrow s i n (2 x) = 2 u = 22 - 8 \sqrt{7} = 0.8340$ $(i t s m e a n [0 \sim 360] : x = 28, 26, 61.74 d e g)$
	Commented by malwan last updated on 23/Jul/24


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