Let α and β respectively be the maximum and the minimum values of the function f(θ)=4 (sin ^4 ( 7 π 2-θ )+sin ^4(11 π+θ) )-2 (sin ^6 ( 3 π 2-θ )+sin…

$$f(\theta) = 4\left\{\sin^4\left(\dfrac{7\pi}{2} - \theta\right) + \sin^4(11\pi + \theta)\right\} - 2\left\{\sin^2\left(\dfrac{3\pi}{2} - \theta\right) + \sin^2(9\pi - \theta)\right\}$$f(θ)=4{sin4(27π​−θ)+sin4(11π+θ)}−2{sin2(23π​−θ)+sin2(9π−θ)}

$$f(\theta) = 4(\cos^4(\theta) + \sin^4(\theta)) - 2(\cos^6\theta + \sin^6\theta)$$f(θ)=4(cos4(θ)+sin4(θ))−2(cos6θ+sin6θ)

$$f(\theta) = 4(1 - 2\sin^2\theta\cos^2\theta) - 2(1 - 3\sin^2\theta\cos^2\theta)$$f(θ)=4(1−2sin2θcos2θ)−2(1−3sin2θcos2θ)

$$f(\theta) = 2 - 2\sin^2\theta\cos^2\theta$$f(θ)=2−2sin2θcos2θ

$$f(\theta) = 2 - \dfrac{\sin^2(2\theta)}{2}$$f(θ)=2−2sin2(2θ)​

$$\alpha = f(\theta)_{max} = 2$$α=f(θ)max​=2

$$\beta = f(\theta)_{min} = \dfrac{3}{2}$$β=f(θ)min​=23​

$$\Rightarrow \alpha + 2\beta = 5$$⇒α+2β=5