Verify the Identity

Question

sin2∅cot∅=2-2sin²∅

Answer 1

sin2∅cot∅ = 2-2sin²∅ given

[sin2∅][cot∅] = separate functions

[sin(∅ + ∅)][ cot∅ ] = sine of double angle = sine of sum of angles

[sin∅cos∅ + cos∅sin∅][ cot∅ ] = sine of sum of angles identity

[2sin∅cos∅] [ cot∅ ] = add terms

[2sin∅cos∅] [ cos∅/sin∅ ] = cotangent as division of cosine over sine

[2sin∅cos∅] [ cos∅/sin∅ ] = simplify

[ 2cos∅ ] [ cos∅ ] = simplify

[ 2cos²∅ ] = simplify

[ 2(1 - sin²∅)] = Pythagorean identity

2 - 2sin²∅ = 2 - 2sin²∅ distrubute

Answer 2

Use the double angle formula to simplify the left hand side

sin(2Φ)cot(Φ) = 2sin(Φ)cos(Φ)(cos(Φ)/sin(Φ))= 2cos(Φ)cos(Φ) = 2cos²(Φ) = 2(1-sin²(Φ) = 2-2sin²(Φ)

QED

Answer 3

sin2∅cot∅=2-2sin²∅

∅=30°

sin60*cos30/sin30=2-2*(sin30)^2

.866*1.732 =1.5