If \cos^2 \theta - \sin^2 \theta - 3 \cos \theta + 2 =0, 0^\circ < \theta < 90^\circ, then what is the value of 4 \cosec \theta + \cot \theta ?

Question 69

If $$\cos^2 \theta - \sin^2 \theta - 3 \cos \theta + 2 =0, 0^\circ < \theta < 90^\circ$$, then what is the value of $$4 \cosec \theta + \cot \theta$$ ?

Solution

We know that :
$$\sin^2\theta\ =1-\cos^2\theta\ \ $$

We can write :

$$\cos^2\theta\ +\left(1-\sin^2\theta\ \right)-3\cos\theta\ +1$$
$$2\cos^2\theta\ -3\cos\theta\ +1\ =0\ $$
On solving ,we get

$$\cos\theta=\frac{1}{2}\ $$
or $$\theta=60$$
so $$4\operatorname{cosec}60+\cot60$$

$$4\times\ \frac{2}{\sqrt{\ 3}}+\frac{1}{\sqrt{\ 3}}$$

= $$\frac{9}{\sqrt{\ 3}}$$

=$$3\sqrt{\ 3}$$

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