Question

Verify the identity by using trigonometric identities:

csc^2(x) - 2csc(x)cot(x) + cot^2 = tan^2(x/2)

Answer 1

Working with the right hand side (RHS):

Recall the half-angle identities,

sin²(x/2) = (1 - cos(x))/2

cos²(x/2) = (1 + cos(x))/2

so that

RHS = tan²(x/2)

RHS = sin²(x/2) / cos²(x/2)

RHS = (1 - cos(x)) / (1 + cos(x))

Multiply the numerator and denominator by 1 - cos(x).

RHS = (1 - cos(x)) / (1 + cos(x)) • (1 - cos(x)) / (1 - cos(x))

RHS = (1 - cos(x))² / (1 - cos²(x))

Recall that

sin²(x) + cos²(x) = 1

so that

RHS = (1 - cos(x))² / (1 - cos²(x))

RHS = (1 - cos(x))² / sin²(x)

Expanding the numerator yields

RHS = (1 - 2 cos(x) + cos²(x)) / sin²(x)

RHS = 1/sin²(x) - 2 cos(x)/sin²(x) + cos²(x)/sin²(x)

RHS = 1/sin²(x) - 2 cos(x)/sin(x) • 1/sin(x) + cos²(x)/sin²(x)

and by definition of csc and cot,

RHS = csc²(x) - 2 cot(x) csc(x) + cot²(x)

as required.