Question

Prove that {(cos^3 theta + sin^3theta)} /(cos theta + sin theta) + {(cos^3 theta - sin^3theta)} /(cos theta - sin theta)=2

Answer 1

Solution,

LHS = $\frac{cos^3\theta+sin^3\theta }{cos\theta+sin\theta} +\frac{cos^3\theta-sin^3\theta }{cos\theta-sin\theta}$

       =$\frac{(cos\theta+sin\theta)(cos^2\theta-cos\theta.sin\theta+sin^2\theta)}{cos\theta+sin\theta} +\frac{(cos\theta-sin\theta)(cos^2\theta+cos\theta.sin\theta+sin^2\theta)}{cos\theta-sin\theta}$  

       =(${cos^2\theta-cos\theta.sin\theta+sin^2\theta)} +{cos^2\theta+cos\theta.sin\theta+sin^2\theta)}$

      = $sin^2\theta+cos^2\theta+sin^2\theta+cos^2\theta+sin\theta.cos\theta-sin\theta.cos\theta$

      = 1+1+0

      =2

      =RHS proved...

Hope it helped.