Question

prove that : (sintheta/1-costheta-1-costheta/sintheta)×(costheta/1-sintheta-1-sintheta/costheta) = 4​

Answer 1

Solution:

Theta is replaced by A.

Used the Trigonometric Identity, as well as other Algebraic identity

Sin²A+Cos²A=1

a²-b²=(a-b)(a+b)

LHS

$=[\frac{sinA}{1-CosA}-\frac{1-Cos A}{SinA}]\times [\frac{CosA}{1-SinA}-\frac{1-SinA}{CosA}]\\\\= [\frac{Sin^2A-(1-CosA)^2}{SinA(1-CosA)}]\times [\frac{Cos^2A-(1-SinA)^2}{CosA(1-SinA)}]\\\\= [\frac{(1-Cos^2A)-(1-CosA)^2}{SinA(1-CosA)}]\times [\frac{(1-Sin^2A)-(1-SinA)^2}{CosA(1-SinA)}]\\\\= (1-CosA)\times(1-SinA)\times [\frac{(1+CosA)-(1-CosA)}{SinA(1-CosA)}]\times [\frac{(1+SinA)-(1-SinA)}{CosA(1-SinA)}]\\\\=\frac{2CosA*2SinA}{SinA*CosA}\\\\=4$

=RHS

Hence Proved.