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  • Prove that sq root sec power 2 theta + cosec power 2 theta = tan theta + cot theta

Prove that \sqrt{\sec ^{2}\theta +\cos ec^{2}\theta }= \tan \theta +\cot \theta

Answers (1)

Taking left-hand side
  \sqrt{\sec ^{2}\theta +\cos ec^{2}\theta }
= \sqrt{\frac{1}{\cos ^{2}\theta }+\frac{1}{\sin ^{2}\theta }}         \left [ \because \sec \theta = \frac{1}{\cos \theta } ,\cos ec= \frac{1}{\sin \theta }\right ]
= \sqrt{\frac{\sin ^{2}\theta +\cos ^{2}\theta }{\cos ^{2}\theta \cdot \sin ^{2}\theta }}
= \sqrt{\frac{1}{\cos ^{2}\theta\cdot \sin ^{2} \theta }}               \left [ \because \sin ^{2}\theta +\cos ^{2}\theta = 1 \right ]
\frac{1}{\cos \theta \sin \theta }=\frac{sin^2\theta+cos^2\theta}{cos \theta sin\theta}=\frac{sin^2\theta}{cos \theta sin\theta}+\frac{cos^2\theta}{cos \theta sin\theta}\\\\=\frac{sin\theta}{cos\theta}+\frac{cos\theta}{sin\theta}=tan\theta+cot\theta=RHS\\\text{Hence proved} 

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infoexpert27

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