Question

Write cos x in terms of cot x

Answer 1

first, use this identity
csc^2 (x) = (cot^2 x + 1)
and we knowed that csc = 1/sin

so, the identity above can be written as
1/sin^2 (x) = (cot^2 x + 1) or
sin^2 (x) = 1/(cot^2 x + 1)

then use the identity :
sin^2 (x) = 1 - cos^2 (x)

now, we get
sin^2 (x) = 1/(cot^2 x + 1)
1 - cos^2 x = 1/(cot^2 x + 1) or
cos^2 x = 1 - 1/(cot^2 x + 1) (simplied again)
cos^2 x = (cot^2 x + 1 - 1)/(cot^2 x + 1)
cos^2 x = (cot^2 x)/(cot^2 x + 1)
and finally, we get
cos(x) = sqrt[(cot^2 x)/(cot^2 x + 1)]