Question

establish the identity 1-(1/2)sin(2θ)=(sin^3 θ +cos^3 θ)/(sinθ+cosθ)

Answer 1

A few things to know 1st

\[\sin^2(\theta) + \cos^2(\theta) = 1\]

\[\sin(2\theta) = 2\sin(\theta)\cos(\theta)\]

sin^3 + cos^3 is the sum of 2 cubics

so

\[\sin^3(\theta)+\cos^3(\theta) = (\sin(\theta) + \cos(\theta))(\sin^2 \theta - \sin(\theta)\cos(\theta) + \cos^2(\theta))\]

after substituting you get

\[1 - \frac{1}{2} \times (2\sin(\theta)\cos(\theta)) = \frac{(\sin(\theta) + \cos(\theta))(\sin^2(\theta) + \sin^2(\theta) - \sin(\theta)(\cos(\theta))}{(\sin(\theta)\cos(\theta))}\]


look to simplify the right hand side and you'll get the simplified left hand side.

hope this helps