Question

Verify the identity

tan 0-cot 0 / tan 0 + cot 0 (not on the same line) + 2cos^2 0=1

Answer 1

See below for the proof of the identity

Explanation:

I read your statement as
`\displaystyle (\tan\theta-\cot\theta)/(\tan\theta+\cot\theta)+2\cos^2\theta=1`:

`\displaystyle (\tan\theta-\cot\theta)/(\tan\theta+\cot\theta)+2\cos^2\theta\\\\((\sin\theta)/(\cos\theta)-(\cos\theta)/(\sin\theta))/((\sin\theta)/(\cos\theta)+(\cos\theta)/(\sin\theta)) +2\cos^2\theta\\\\((\sin^2\theta-\cos^2\theta)/(\sin\theta\cos\theta))/((\sin^2\theta+\cos^2\theta)/(\sin\theta\cos\theta)) +2\cos^2\theta\\\\((\sin^2\theta-\cos^2\theta)/(\sin\theta\cos\theta))/((1)/(\sin\theta\cos\theta)) +2\cos^2\theta`

`\displaystyle \sin^2\theta-\cos^2\theta+2\cos^2\theta\\\\\sin^2\theta+\cos^2\theta\\\\1`

Thus, the identity is proven.