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If cosec theta + cot theta = K, then prove that cos theta = K^2 -1/K^2 +1.

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$\\cosec\theta+cot\theta=k.......(1) \\cosec^2\theta-cot^2\theta=1\\(cosec\theta-cot\theta)(cosec\theta+cot\theta)=1\\cosec\theta-cot\theta=\frac{1}{k}........(2)\text{from (1) and (2)}\\2cot\theta=k-\frac{1}{k}\\2coosec\theta=k+\frac{1}{k}\\\\\frac{2cot\theta}{2cosec\theta}=cos\theta=\frac{k^2-1}{k^2+1}$

Posted by

Safeer PP

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Cosec theta+cos theta=k

1/sin theta+cos theta/sin theta=k

1+cos theta/sin theta =k

Sqaring on both sides

(1+cos theta/sin theta)²=(k)²

(1+cos theta)²/sin² theta=k²

(1+cos theta)(1+cos theta)/ sin² theta=k²

(1+cos theta)(1+cos theta)/(1²-cos² theta)=k²

(1+cos theta)(1+cos theta)/(1+cos theta)(1-cos theta)=k²

1+cos theta/1-cos theta=k²

1+cos theta=k²(1-cos theta)

1+Cos theta=k²-k²cos theta

Cos theta+k²xos theta =k²-1

Cos theta(1+k²)=k²-1

Cos theta=k²-1/k²+1

Hence proved

Posted by

Pspk

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