Question

Ln(1-cosx)+ln(1+cosx) =2lncosx

Answer 1

Use log rules to combine left side:
$ln (1-cos x)(1+cos x) = 2 ln (cos x) \\ \\ ln (1-cos^2 x) = 2 ln (cos x)$

Use pythagorean trig identity: sin^2 = 1-cos^2

$ln (sin^2 x) = 2 ln (cos x) \\ \\ 2ln(sin x) = 2 ln (cos x) \\ \\ ln (sin x) = ln (cos x)$

Take e raised to both sides, this will cancel out the log.
$sin x = cos x \\ \\ \frac{sin x}{cos x} = 1 \\ \\ tan x = 1 \\ \\ x = \frac{\pi}{4}$