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Question Number 11844 by minakshidahaval0202@gmail.co. last updated on 02/Apr/17

$$\cot \alpha +\mathrm{cosec}\alpha =kthenfind\mathrm{cosec}\alpha -cot\alpha andalsofindcot\alpha$$

Answered by sma3l2996 last updated on 02/Apr/17

$$wehave$$$$\left(i\right):cot\alpha +cosec\alpha =\frac{cos\alpha }{sin\alpha }+\frac{1}{sin\alpha }=k$$$$=\frac{cos\alpha +1}{sin\alpha }=\frac{(cos\alpha +1)(cos\alpha -1)}{sin\alpha (cos\alpha -1)}=\frac{-{sin}^2\alpha }{sin\alpha (cos\alpha -1)}$$$$=\frac{1}{\frac{1}{sin\alpha }-\frac{cos\alpha }{sin\alpha }}=\frac{1}{cosec\alpha -cot\alpha }=k$$$$\left(ii\right):cosec\alpha -cot\alpha =\frac{1}{k}$$$$letdo$$$$\left(i\right)-\left(ii\right):2cot\alpha =k-\frac{1}{k}$$$$cot\alpha =\frac{k^2-1}{k}$$

Commented by sma3l2996 last updated on 02/Apr/17

$$Imeancot\alpha =\frac{k^2-1}{2k}$$

Answered by ajfour last updated on 02/Apr/17

$$\mathrm{cosec}{}^2\alpha -\cot {}^2\alpha =1$$$$(\mathrm{cosec}\alpha -\cot \alpha )(\mathrm{cosec}\alpha +\cot \alpha )=1$$$$(\mathrm{cosec}\alpha -\cot \alpha )\left(k\right)=1$$$$\mathrm{cosec}\alpha -\cot \alpha =\frac{1}{k}...\left(ii\right)$$$$andas\mathrm{cosec}\alpha +\cot \alpha =k...\left(i\right)$$$$\left(i\right)-\left(ii\right)gives$$$$2\cot \alpha =k-\frac{1}{k}$$$$\cot \alpha =\frac{k^2-1}{2k}.$$

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