Prove the identity tanx cotx 2 csc2x Prove the identity 1

Prove the identity: tan(x) + cot(x) = 2 csc(2x). Prove the identity: 1 + cot^2(x) = csc^2(x). Solve for x: 4cos(x)sin(x) + 2cos(x) + 2sin(x) + 1 = 0. Solve for x: sin^2(x + 1) = cos^2(x + 1)

Solution

7) tanx +cotx = 2csc(2x)

LHS : tanx +1/tanx = (1 +tan^2x)/tanx

sinx/cosx + cosx/sinx

= (sin^2x +cos^2x)/sinx*cosx

= 2(sin^2x +cos^2x)/2sinx*cosx

= 2/sin2x

= 2csc2x

RHS

8) 1 + cot^2x = csc^2x

LHS : 1 + cos^2x/sin^2x

( sin^2x +cos^2x)/sin^2x

= 1/sin^2x

= csc^2x

RHS