Question

if cosec theta+cot theta=K,prove that cos theta =k²- 1/k²+1​

Answer 1

Step-by-step explanation:

$cosec\theta + cot\theta = k\\\\\frac{1}{sin\theta} + \frac{cos\theta}{sin\theta} = k\\\\1+ cos\theta = ksin\theta\\\\1+cos\theta = k\sqrt{1-cos^{2}\theta}\\\\1+ cos^{2}\theta + 2cos\theta = k^{2}(1-cos^{2}\theta)\\\\1+ cos^{2}\theta + 2cos\theta + k^{2}cos^{2}\theta -k^{2} = 0$

Solve This quadratic equation to get values of cosθ

$cos^{2}\theta(k^{2}+1) + 2cos\theta + 1- k^{2} = 0\\\\d = 4 - 4(k^{2} + 1)(1-k^{2})\\\\d = 4(1-(1-k^{4}))\\\\d = 4k^{4}\\\\roots = \frac{-b\pm\sqrt{D}}{2a}\\\\roots = \frac{-2\pm2k^{2}}{2(k^{2}+1)}\\\\root_{1} = \frac{-2(1+k^{2})}{2(k^{2}+1)}\\\\root_{1} = -1\\\\root_{2} = \frac{-2(1-k^{2})}{2(k^{2}+1)}\\\\root_{2}= \frac{k^{2}-1}{k^{2}+1}$

Now cosθ = -1 or cosθ = $\frac{k^{2}-1}{k^{2}+1}$

But if cosθ = -1 Then cotθ and cosecθ are not defined so cosθ = $\frac{k^{2}-1}{k^{2}+1}$

Hence proved also you can put value to verify!