Question

Prove that secA ( 1 - sinA ) ( secA + tanA ) = 1

Answer 1

sec A(1-sin A)(sec A+tan A)
(1-sin A)(1+sin A)/(cos²A)=1

Note: Step 1 Convert secant and tangent in sine and cosine.

Step 2 (a-b)(a+b)=a²-b²

Also, 1-sin²A=cos²A

Answer 2

LHS

$secA(1-sinA)(secA+tanA) \\ \\ = \frac{1}{cosA} (1-sinA)(secA+tanA) \\ \\ = ( \frac{1}{cosA} - \frac{ sinA}{cosA} )(secA+tanA) \\ \\ = (secA-tanA)(secA+tanA) \\ \\ = {sec}^{2} A - {tan}^{2} A\\ \\ = 1$

RHS

=1

LHS=RHS