ᴓ=pheta
1. (sinᴓ + cosᴓ)² = 1+sin2ᴓ
2.cot2ᴓ=cos²ᴓ-sin²ᴓ over 2sinᴓ cosᴓ
3.tan2ᴓ=2 over cotᴓ-tanᴓ
1. Expand like you would any other function:
$\displaystyle sin^2(\theta) + 2sin(\theta)cos(\theta) + cos^2(\theta) = 1+ 2sin(\theta)$
It would appear number 1 is not an identity. (try 90deg)
2. $\displaystyle cot(\theta) = \frac{cos^2(\theta)-sin^2(\theta)}{2sin(\theta)cos(\theta)} = \frac{cos(2\theta)}{sin(2\theta)} = cot(2(\theta))$
2 is not an identity either (try 45 degrees)
3. $\displaystyle \frac{2}{cot(\theta)-tan(\theta)} = \frac{2}{\frac{cos^2(\theta)-sin^2(\theta)}{cos(\theta)sin(\theta)}}$
Looking at the denominator: $\displaystyle cos(\theta)sin(\theta) = \frac{1}{2}sin(2\theta)$
$\displaystyle \frac{cos^2(\theta)-sin^2(\theta)}{cos(\theta)sin(\theta)} = \frac{2cos(2\theta)}{sin(2\theta} = 2cot(2\theta)$
The original expression becomes: $\displaystyle \frac{2}{2cot(2\theta)} = tan(2\theta)$