Question

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

Answer 1

according to question lhs are equal to rhs

Answer 2

TO PROVE THAT :-

$\sf \sqrt{\dfrac{(1+sinA)}{(1-sinA)}} = secA+tanA$

SOLUTION :-

$\sf L.H.S = \sqrt{\dfrac{(1+sinA}{(1-sinA)}}$

Multiply the numerator and the denominator by $\sf (1+sinA)$ .

        $= \sf \sqrt{\dfrac{(1+sinA) \times (1+sinA)}{(1-sinA) \times (1+sinA)}} \\\\= \sqrt{\dfrac{(1+sinA)^2}{(1-sin^2A)}}$

$\bullet\bf \ 1-sin^2A = cos^2A$

        $= \sf \sqrt{\dfrac{(1+sinA)^2}{cos^2A}} \\\\ = \dfrac{1+sinA}{cosA}\\\\= \dfrac{1}{cosA}+\dfrac{sinA}{cosA}$

$\bullet\bf \ \dfrac{1}{cosA}= secA\\\\\bullet \dfrac{sinA}{cosA}= tanA$

        $= \sf secA+tanA \\\\= R.H.S$

$\therefore \sf R.H.S = L.H.S$

Hence proved.