Question

Prove that cosA/(1-sinA)=tan(π/4+A/2)

Answer 1

CosA / 1 – SinA can be written as
Cos²A/2 – Sin²A/2 / Cos²A/2 + Sin²A/2 +2Sin A/2 Cos A/2 ….…
which is equal to = Cos²A/2 – Sin²A/2 / (CosA/2 + SinA/2)² .. As Sin²A/2 + Cos²A/2 + 2Sin A/2 Cos A/2 is in the form of X² + Y² +2XY = (X+Y)²
And Cos²A/2 – Sin²A/2 can be written as (CosA/2 – SinA/2) (CosA/2 + SinA/2) .. as it is in the form of (X + Y)(X – Y) = X² – Y²
So Cos²A/2 – Sin²A/2 / Cos²A/2 + Sin²A/2 +2Sin A/2 Cos A/2 =
(CosA/2 – SinA/2) (CosA/2 + SinA/2) / (CosA/2 + SinA/2)2 =
(CosA/2 – SinA/2) (CosA/2 + SinA/2) / (CosA/2 + SinA/2)2 =
(CosA/2 – SinA/2) / (CosA/2 + SinA/2) .
Now divide the numerator and the denominator by CosA/2 .. because of which we get ..
1 – tan A/2 / 1 + tan A/2 = tan 45 + tan A/2 / 1 – tan 45 * tan A/2 = tan (45 +A/2) =tan(π/4+A/2)
Guess it helped . :)
Revised Answer