Prove the following trigonometric identities.

Question:

Prove the following trigonometric identities.

$(\operatorname{cosec} A-\sin A)(\sec A-\cos A)(\tan A+\cot A)=1$

Solution:

We have to prove $(\operatorname{cosec} A-\sin A)(\sec A-\cos A)(\tan A+\cot A)=1$

We know that, $\sin ^{2} A+\cos ^{2} A=1$

So,

$(\operatorname{cosec} A-\sin A)(\sec A-\cos A)(\tan A+\cot A)$

$=\left(\frac{1}{\sin A}-\sin A\right)\left(\frac{1}{\cos A}-\cos A\right)\left(\frac{\sin A}{\cos A}+\frac{\cos A}{\sin A}\right)$

$=\left(\frac{1-\sin ^{2} A}{\sin A}\right)\left(\frac{1-\cos ^{2} A}{\cos A}\right)\left(\frac{\sin ^{2} A+\cos ^{2} A}{\sin A \cos A}\right)$

$=\left(\frac{\cos ^{2} A}{\sin A}\right)\left(\frac{\sin ^{2} A}{\cos A}\right)\left(\frac{1}{\sin A \cos A}\right)$

$=\frac{\sin ^{2} A \cos ^{2} A}{\sin ^{2} A \cos ^{2} A}$

$=1$

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