Question

Prove that
{(cosA/1+sinA)+(1+sinA/cosA)}×{(cosA/1-sinA)-(1-sinA/cosA)}

Answer 1

$\bold{Given : \{\frac{CosA}{1 + SinA} + \frac{1 + SinA}{CosA}\}.\{\frac{CosA}{1 - SinA} - \frac{1 - SinA}{CosA}\}}$

$Taking\;LCM,\;We\;Get\;:$

$\implies \{\frac{Cos^2A + (1 + SinA)^2}{CosA(1 + SinA)}\}.\{\frac{Cos^2A - (1 - SinA)^2}{CosA(1 - SinA)}\}$

$\implies \{\frac{Cos^2A + 1 + Sin^2A + 2SinA}{CosA(1 + SinA)}\}.\{\frac{Cos^2A - 1 - Sin^2A + 2SinA}{CosA(1 - SinA)}\}$

$\implies \{\frac{2 + 2SinA}{CosA(1 + SinA)}\}.\{\frac{1 - Sin^2A - 1 - Sin^2A + 2SinA}{CosA(1 - SinA)}\}$

$\implies \{\frac{2(1 + SinA)}{CosA(1 + SinA)}\}.\{\frac{2SinA - 2Sin^2A}{CosA(1 - SinA)}\}$

$\implies \{\frac{2}{CosA}\}.\{\frac{2SinA(1 - SinA)}{CosA(1 - SinA)}\}$

$\bold{\implies \{2 \times SecA\}\{2 \times TanA\}}$