Evaluate the value of the expression: 3(sin θ − cos θ)^4 + 6(sin θ + cos θ)^2 + 4(sin^6 θ + cos^6 θ).

Solution

Solution: Let \(x=\sin\theta\cos\theta\). Since \(\sin^2\theta+\cos^2\theta=1\), we have \((\sin\theta-\cos\theta)^2=1-2x\), so the fourth power is \((1-2x)^2\). Also, \((\sin\theta+\cos\theta)^2=1+2x\), and \(\sin^6\theta+\cos^6\theta=1-3x^2\). The expression becomes \(3(1-2x)^2+6(1+2x)+4(1-3x^2)=13\). Trick: Substitute \(x=\sin\theta\cos\theta\) and use \(\sin^2\theta+\cos^2\theta=1\) to simplify the higher powers.