If _1 and _2 be respectively the smallest and the largest values of in (0,2)- which satisf…

Question

If $	heta_1$ and $	heta_2$ be respectively the smallest and the largest values of $	heta$ in $(0,2\pi)-\{\pi\}$ which satisfy the equation $2\cot^2	heta-\dfrac{5}{\sin	heta}+4=0$, then $\displaystyle\int_{	heta_1}^{	heta_2}\cos^23	heta\,d	heta$ is equal to:

Accepted Answer

$$2\cot^2\theta-\frac5{\sin\theta}+4=0\Rightarrow\frac{2\cos^2\theta}{\sin^2\theta}-\frac5{\sin\theta}+4=0$$

Multiplying by $\sin^2\theta$: $2\cos^2\theta-5\sin\theta+4\sin^2\theta=0\Rightarrow2\sin^2\theta-5\sin\theta+2=0$ (using $\cos^2\theta=1-\sin^2\theta$)
$$\Rightarrow(2\sin\theta-1)(\sin\theta-2)=0\Rightarrow\sin\theta=\frac12\Rightarrow\theta=\frac\pi6,\frac{5\pi}6$$

So $\theta_1=\dfrac\pi6,\ \theta_2=\dfrac{5\pi}6$.
$$\int_{\pi/6}^{5\pi/6}\cos^23\theta\,d\theta=\int_{\pi/6}^{5\pi/6}\frac{1+\cos6\theta}{2}d\theta=\frac12\left[\theta+\frac{\sin6\theta}6\right]_{\pi/6}^{5\pi/6}=\frac12\left(\frac{5\pi}6-\frac\pi6\right)=\frac12\cdot\frac{4\pi}6=\frac\pi3$$

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