sine and cosine

Question

Express the following functions in the form f ◦ g where f is a rational function and g is a trigonometric function:

a) u(t) =cos(t)/ 1+cos(t) .

 b) w(t)=cos(t)/ sin2(t).

Answer 1

a). Let g(t) = cost and f(t) = t / (1 + t)

Then, f(g(t)) = f(cost) = cost / (1 + cost) = u(t)

b). Since sin²t + cos²t = 1, sin²t = 1 - cos²t. So, w(t) can be rewritten as cost / (1 - cos²t).

Let g(t) = cost and f(t) = t / (1 - t²).

Then f(g(t)) = f(cost) = cost / (1 - cos²t) = w(t)