Prove that sec A (1- sin A)(sec A + tan A) = 1

Q) Prove that sec A (1- sin A)(sec A + tan A) = 1

Ans: Let’s start from LHS

LHS = sec A (1- sin A)(sec A + tan A)

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

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

= $\frac{1 - \sin^2A}{\cos^2A}$

We know that sin²A + cos²A = 1

or we can say that, 1 – sin²A = cos²A

$\therefore$ LHS = $\frac{\cos^2A}{\cos^2A}$

= 1 ……….. RHS

Hence proved!

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