Question

Using t=tanx/2, write an expression for sinx and cosx in terms of t.

Answer 1

Assume 0 < x/2 < π/2. Then

tan²(x/2) + 1 = sec²(x/2)   ===>   sec(x/2) = √(1 - tan²(x/2))

===>   cos(x/2) = 1/√(1 - tan²(x/2))

===>   cos(x/2) = 1/√(1 - t ²)

We also know that

sin²(x/2) + cos²(x/2) = 1   ===>   sin(x/2) = √(1 - cos²(x/2))

Recall the double angle identities:

cos(x) = 2 cos²(x/2) - 1

sin(x) = 2 sin(x/2) cos(x/2)

Then

cos(x) = 2/(1 - t ²) - 1 = (1 + t ²)/(1 - t ²)

sin(x) = 2 √(1 - 1/(1 - t ²)) / √(1 - t ²) = 2t/(1 - t ²)