Question

it's trigonometry sum very easy plz try to solve it

Answer 1

here your ans if its help mark me as a brilliant

Answer 2

Here, I am writing θ as A for better understanding!

$Given:\frac{cotA + cosecA - 1}{cotA - cosecA + 1}$

$=>\frac{ \frac{cosA}{sinA}+\frac{1}{sinA}-1}{\frac{cosA}{sinA}-\frac{1}{sinA}+1}$

$=>\frac{\frac{cosA+1-sinA}{sinA}}{\frac{cosA-1 +sinA}{sinA}}$

$=>\frac{cosA+1-sinA}{cosA-1+sinA}$

cosθ = 2 cos²(θ/2) - 1, sinθ = 2 sin(θ/2) cos (θ/2), cos θ = 1 - 2sin²(θ/2).

$=>\frac{2cos^2\frac{A}{2}-1-2sin\frac{A}{2}cos\frac{A}{2}+1}{1-2sin^2\frac{A}{2}+2sin\frac{A}{2}cos\frac{A}{2}-1}$

$=>\frac{2cos^2\frac{A}{2}-2sin\frac{A}{2}cos\frac{A}{2}}{2sin\frac{A}{2}cos\frac{A}{2}-2sin^2\frac{A}{2}}$

$=>\frac{2(cos^2\frac{A}{2}-sin\frac{A}{2}cos\frac{A}{2})}{2(sin\frac{A}{2}cos\frac{A}{2}-sin^2\frac{A}{2})}$

$=>\frac{cos^2\frac{A}{2}-sin\frac{A}{2}cos\frac{A}{2}}{sin\frac{A}{2}cos\frac{A}{2}-sin^2\frac{A}{2}}$

$=>\frac{cos\frac{A}{2}(cos\frac{A}{2}-sin\frac{A}{2})}{sin\frac{A}{2}(cos\frac{A}{2}-sin\frac{A}{2})}$

$=>\frac{cos\frac{A}{2}}{sin\frac{A}{2}}$

$=>\boxed{cot\frac{A}{2}}$

Hope this helps!