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Question Number 8310 by lepan last updated on 07/Oct/16

$$GiventhatcosecA+cotA=3evaluate$$$$cosecA-cotAandcosA.$$$$$$

Answered by prakash jain last updated on 08/Oct/16

$$1+{\cot }^{2}A={\mathrm{cosec}}^{2}A$$$$\Rightarrow {\mathrm{cosec}}^{2}A-{\cot }^{2}A=1$$$$\Rightarrow (\mathrm{cosec}A+\cot A)(\mathrm{cosec}A-\cot A)=1$$$$Given\mathrm{cosec}A+\cot A=3$$$$\Rightarrow 3(\mathrm{cosec}A-\cot A)=1$$$$\Rightarrow (\mathrm{cosec}A-\cot A)=\frac{1}{3}$$$$--------$$$$\mathrm{cosec}A+\cot A=3$$$$(\mathrm{cosec}A-\cot A)=\frac{1}{3}$$$$2\mathrm{cosec}A=3+\frac{1}{3}\Rightarrow \mathrm{cosec}A=\frac{5}{3}$$$$\cot A=\frac{4}{3}$$$$\sin A=\frac{3}{5}$$$$\cos A=\pm \sqrt{1-{\sin }^{2}A}=\pm \frac{4}{5}$$$$\mathrm{since}\cot A=\frac{4}{3}>0$$$$\mathrm{Only}\mathrm{valid}\mathrm{solution}\mathrm{for}\cos A=\frac{4}{5}$$

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