Please help! cot^2 xsin^2 x=cos 2x+sin^2 x AND tan x/ 2sin xcos x = 1/1+cos 2 x

Question

I need help with both. Thank You

Answer 1

1.

LHS = cot²(x) sin²(x) =

(cos²(x)/sin²(x)) sin²(x) =

cos²(x)

RHS = cos2x + sin²(x) =

2cos²(x) - 1 + sin²(x) =

2cos²(x) - cos²(x) =

cos²(x)

Therefore LHS = RHS

2.

LHS = tan(x) / (2sin(x)cos(x)) =

(sin(x)/cos(x)) / (2sin(x)cos(x)) =

1 / (2cos²(x))

RHS = 1 / (1 + cos2x) =

1 / (1 + 2cos²(x) - 1) =

1 / (2cos²(x))

Therefore LHS = RHS