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Question Number 116483 by bemath last updated on 04/Oct/20

$$\mathrm{If}8\cos \mathrm{x}-8\sin \mathrm{x}=3,\mathrm{what}\mathrm{is}\mathrm{the}$$$$\mathrm{value}\mathrm{of}55\tan \mathrm{x}+\frac{55}{\tan \mathrm{x}}?$$

Answered by mr W last updated on 04/Oct/20

$$8(1-\tan x)=\frac{3}{\cos x}$$$$64{(1-\tan x)}^2=9(1+{\tan }^{2}x)$$$$64-128\tan x+64{\tan }^{2}x=9+9{\tan }^{2}x$$$$55-128\tan x+55{\tan }^{2}x=0$$$$\frac{55}{\tan x}-128+55\tan x=0$$$$\Rightarrow 55\tan x+\frac{55}{\tan x}=128$$

Answered by bobhans last updated on 04/Oct/20

$$\left(1\right)\cos \mathrm{x}-\sin \mathrm{x}=\frac{3}{8}$$$$\Rightarrow 1-\sin 2\mathrm{x}=\frac{9}{64},\sin 2\mathrm{x}=\frac{55}{64}$$$$\Rightarrow \tan \mathrm{x}=\frac{1-\cos 2\mathrm{x}}{\sin 2\mathrm{x}}=\frac{1-\sqrt{1-\sin {}^{2}2\mathrm{x}}}{\sin 2\mathrm{x}}$$$$\Rightarrow \tan \mathrm{x}=\frac{1-\sqrt{(1+\frac{55}{64})(1-\frac{55}{64})}}{\frac{55}{64}}$$$$\Rightarrow 55\tan \mathrm{x}=64\{1-\sqrt{\frac{119}{64}\times \frac{9}{64}}\}$$$$\Rightarrow 55\tan \mathrm{x}=64-3\sqrt{119}$$$$\mathrm{and}55\tan \mathrm{x}+\frac{55}{\tan \mathrm{x}}=55(\tan \mathrm{x}+\frac{1}{\tan \mathrm{x}})$$$$=55(\frac{64-3\sqrt{119}}{55}+\frac{1}{64-3\sqrt{119}})$$$$=55\left(\frac{5222-384\sqrt{119}}{55\times (64-3\sqrt{119}}\right)=\frac{5222-384\sqrt{119}}{64-3\sqrt{119}}$$

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