Question

Prove the following identities: `(sin^3theta+cos^3theta)/(sintheta+costheta)+sinthetacostheta=1`

Answer 1

We have
` "LHS " = (sin^(3) theta + cos ^(3) theta)/(sin theta+ cos theta )+ sin theta cos theta `
` = ((sin theta+ cos theta)(sin^(2) theta+ cos ^(2) theta- sin theta cos theta))/((sin theta + cos theta ))+ sin theta cos theta `
` [ because (a^(3)+b^(3))=(a+b)(a^(2)+b^(2)-ab)] `
` =(1- sin theta cos theta)+ sin theta cos theta=1 " " [ because sin^(2) theta+ cos^(2) theta =1] `
`= "RHS. "`
` therefore " LHS "= "RHS. " `