Hello! Can anyone help me verify the following? (1+cos(2y))/sin(2y)=cot(y)

Question

So far I have separated the left hand side into two fractions so that I have 1/sin2y + cos2y/sin2y and then was able to simplify 1/sin2y to csc2y by the reciprocal identity. But that is as far as I could get before I got confused. Thank you for the help in advance!

Answer 1

Wrong approach. First use the double angle identities to get rid of the "2y" in the numerator and denominator of the left side:

cos(2y) = cos²(y) - sin²(y)

sin(2y) = 2sin(y)cos(y)

That makes the left side:

(1 + cos²(y) - sin²(y))/(2sin(y)cos(y))

(1 - sin²(y) + cos²(y))/(2sin(y)cos(y))

Then use the Pythagorean Identity sin²(y) + cos²(y) = 1 except flip it to cos²(y) = 1 - sin²(y)

(cos²(y) + cos²(y))/(2sin(y)cos(y))

2cos²(y)/(2sin(y)cos(y))

cos(y)/sin(y)

cot(y)