Question

If cosx = tanx, determine all possible values of sinx.

in not sure how to do this question. I just wrote sinx = cos^2x but I think there is more to this problem.

any ideas

tnx a million

Answer 1

If
`\cos x=\tan x`, then


`\cos x=(\sin x)/(\cos x)\implies\cos^2x=\sin x`

for
`\cos x\\eq0`. This is definitely the case, since
`\tan x` is undefined if
`\cos x=0`. Now,
`\cos^2x=1-\sin^2x`, so we have


`1-\sin^2x=\sin x\implies\sin^2x+\sin x-1=0\implies\sin x=\frac{-1\pm\sqrt5}2`

or
`x\approx-1.618` and
`x\approx0.618`. One of these solutions is larger than 1 in absolute value, but
`|\sin x|\le1`, so we omit that solution, leaving us with


`\sin x=\frac{\sqrt5-1}2\implies x=\arcsin\left(\frac{\sqrt5-1}2\right)+2n\pi`

for integers
`n`, which follows from the fact that
`\sin x` has period
`2\pi`.