Question

Evaluate:
$(1+\cos x)dy=(1-\cos x)dx$

Answer 1

$(1+\cos x)dy=(1-\cos x)dx$

$\int dy=\int \frac{1-\cos x}{1+\cos x}dx$

$\cos x=1-2{\sin }^2\left(x/2\right)$

$\cos x=2{\cos }^2\left(x/2\right)-1$

$y=\int \frac{1-(1-2{\sin }^2(x/2\left)\right)}{1+2{\cos }^2\left(x/2\right)-1}dx=\int \frac{{\sin }^2\left(x/2\right)}{{\cos }^2\left(x/2\right)}dx$

$y=\int {\tan }^{2}xdx=\int ({\sec }^{2}x-1)dx$ $[\because {\sec }^{2}x-{\tan }^{2}x=1]$

$y=\tan x-x+c$.