Question

prove that secA(1-sinA)(secA + tanA)=1​

Answer 1

To Prove :

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

Proof :

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

LHS

» sec A(1 - sin A)(sec A + tan A)

» (sec A - sec A sin A)(sec A + tan A)

» (sec A - (1/cos A) sin A)(sec A + tan A)

» (sec A - sin A/cos A)(sec A + tan A)

» (sec A - tan A)(sec A + tan A)

» sec² A - tan² A

» 1

» RHS

Hence, proved

Formulae applied here :

→ sin A/cos A = tan A

→ (a + b)(a - b) = a² - b²

→ sec² - tan² = 1 (identity used)

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