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

$$\mathrm{If}\sin x+\cos x+\tan x+\mathrm{cosec}x+$$$$\cot x+\sec x=7,\mathrm{then}\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}$$$$\sin 2x.$$

Answered by ajfour last updated on 15/Aug/17

$$\sin \mathrm{x}+\cos \mathrm{x}+\frac{1}{\sin \mathrm{x}}+\frac{1}{\cos \mathrm{x}}+\frac{\sin \mathrm{x}}{\cos \mathrm{x}}$$$$+\frac{\cos \mathrm{x}}{\sin \mathrm{x}}=7$$$$\Rightarrow (\sin \mathrm{x}+\cos \mathrm{x})[1+\frac{2}{\sin 2\mathrm{x}}]+\frac{2}{\sin 2\mathrm{x}}=7$$$$\mathrm{let}\sin 2\mathrm{x}=\mathrm{t}$$$$\mathrm{and}\mathrm{as}{(\sin \mathrm{x}+\cos \mathrm{x})}^2=1+\sin 2\mathrm{x}$$$$\Rightarrow (\sin \mathrm{x}+\cos \mathrm{x})=\sqrt{1+\mathrm{t}}$$$$\Rightarrow \sqrt{1+\mathrm{t}}(1+\frac{2}{\mathrm{t}})+\frac{2}{\mathrm{t}}=7$$$$\Rightarrow \sqrt{1+\mathrm{t}}(1+\frac{2}{\mathrm{t}})+(1+\frac{2}{\mathrm{t}})=8$$$$\Rightarrow (1+\frac{2}{\mathrm{t}})(\sqrt{1+\mathrm{t}}+1)=8$$$$\Rightarrow \left(\frac{\mathrm{t}+2}{\mathrm{t}}\right)(\frac{\mathrm{t}}{\sqrt{1+\mathrm{t}}-1)}=8$$$$\Rightarrow (\mathrm{t}+2)=8\sqrt{1+\mathrm{t}}-8$$$$\Rightarrow 64(1+\mathrm{t})={(\mathrm{t}+10)}^2$$$$\Rightarrow {\mathrm{t}}^2-44\mathrm{t}+36=0$$$$\Rightarrow {(\mathrm{t}-22)}^2=484-36$$$$\Rightarrow \mathrm{t}-22=\pm \sqrt{{\left(21\right)}^2+7}$$$$\Rightarrow \mathrm{So}\sin 2\mathrm{x}=\mathrm{t}=22-\sqrt{{\left(21\right)}^2+7}.$$

Commented by Tinkutara last updated on 15/Aug/17

$$\mathrm{Thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{Sir}!$$

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