Question

Prove that ( − sin )(sec − cos )(tan +
cot) = 1

Answer 1

Step-by-step explanation:

$\sec\theta-\cos\theta=\frac{1}{\cos\theta} -\cos\theta=\frac{1-\cos^2\theta}{\cos\theta} =\frac{\sin^2\theta}{\cos\theta} =\sin\theta\tan\theta\\\\\csc\theta-\sin\theta=\frac{1}{\sin\theta}-\sin\theta=\frac{1-\sin^2\theta}{\sin\theta} =\frac{\cos^2\theta}{\sin\theta}=\cos\theta\cot\theta$$(\csc\theta-\sin\theta)(\sec\theta-\cos\theta)(\tan\theta+\cot\theta)\\=\cos\theta\cot\theta\sin\theta\tan\theta(\tan\theta+\cot\theta)\\=\sin\theta\cos\theta\tan\theta+\frac{\sin\theta\cos\theta}{\tan\theta} \\=\sin^2\theta+\cos^2\theta=1$

Hence proved :)