Question

Prove the following identity: `(1/(sec^2theta-cos^2theta)+1/(cos e c^2theta-sin^2theta))sin^2thetacos^2theta=(1-sin^2thetacos^2theta)/(2+sin^2cos^2theta)`

Answer 1

`LHS={(1)/(((1)/(cos^(2)theta)-cos^(2)theta))+(1)/(((1)/(sin^(2)theta)-sin^(2)theta))}(sin^(2)thetacos^(2)theta)`
`={(cos^(2)theta)/((1-cos^(4)theta))+(sin^(2)theta)/((1-sin^(4)theta))}sin^(2)theta cos^(2)theta`
`={(cos^(2)theta)/(sin^(2)theta(1+cos^(2)theta))+(sin^(2)theta)/(cos^(2)theta(1+sin^(2)theta))}sin^(2)thetacos^(2)theta`
`=(cos^(4)theta(1+sin^(2)theta)+sin^(4)theta(1+cos^(2)theta))/((1+cos^(2)theta)(1+sin^(2)theta))`
`=(cos^(4)theta+sin^(4)theta+cos^(2)thetasin^(2)theta(cos^(2)theta+sin^(2)theta))/(1+(sin^(2)theta+cos^(2)theta)+sin^(2)thetacos^(2)theta)`
`=((cos^(2)theta+sin^(2)theta)^(2)-sin^(2)thetacos^(2)theta)/(2+sin^(2)theta cos^(2)theta)=(1-sin^(2)thetacos^(2)theta)/(2+sin^(2)theta cos^(2)theta)`