how do you solve this.prove: cos2A plus cosA divided by sin2A minus sinA equal to cos2A plus 1 divided by sinA

Answer 1

I think there's a typo in your equation. For example , plug in A = pi/4. You then have

(0 + 1/√2)/(1 - 1/√2) = (0+1)/(1/√2)
1/√2 *√2 /(√2 - 1) = √2
1/(√2-1) = √2
√2+1 = √2

However, if you fix the right side to read (cosA+1)/sinA

you have the left side:

(cos^2 A - sin^2 A + cosA)/(2sinAcosA - sinA)

(cos^2 A - (1 - cos^2 A) + cosA)/(2sinAcosA - sinA)

(2cos^2 A + cosA - 1)/[sinA(2cosA - 1)

(2cosA-1)(cosA - 1)/[(2cosA - 1)(sinA)]

(cosA -1)/sinA

Now if you plug in any angle, the equality holds.