Write the trigonometric expression below in terms of sine and cosine. Simplify if possible.

Question

1) sec(x)cot(x)

2) (cot(t) - csc(t))(cot(t) + csc(t))

Answer 1

1) sec(x)cot(x) = 1/cos(x) · cos(x)/sin(x). The cosines cancel and we are left with 1/sin(x) which is csc(x).

2) (cot(t) - csc(t))(cot(t) + csc(t)) = (cos(t)/sin(t) - 1/sin(t))(cos(t)/sin(t) + 1/sin(t)) = [(cos(t)-1)/sin(t)][(cos(t)+1)/sin(t)] = (cos²t - 1)/sin²t = -sin²(t)/sin²(t) = -1.

(Problem #2 is easier to do without writing as sines and cosines: (cot(t) - csc(t))(cot(t) + csc(t)) = cot²(t) - csc^t(t) = -1 by an identity that relates cot and csc.)