Question

Use trigonometric identities to verify each expression is equal. csc2(x) - 2csc(x)cot(x) + cot2(x) = tan2(x/2)

Answer 1

By definition of csc and cot,

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

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

Combining fractions,

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

Factorize the numerator and use the Pythagorean identity,

cos²(x) + sin²(x) = 1,

to rewrite the denominator.

… = (1 - cos(x))² / (1 - cos²(x))

Cancel a factor of 1 - cos(x).

… = (1 - cos(x)) / (1 + cos(x))

Using the half-angle identities,

sin²(x) = (1 - cos(2x))/2

cos²(x) = (1 + cos(2x))/2

it follows that

1 - cos(x) = 2 sin²(x/2)

1 + cos(x) = 2 cos²(x/2)

and so

… = (2 sin²(x/2)) / (2 cos²(x/2))

… = sin²(x/2) / cos²(x/2)

Finally, by definition of tangent,

… = tan²(x/2)