Trigonometric EquationHard

Question

The least value of $\left( \cos^{2}\theta - 6sin\theta cos\theta + 3\sin^{2}\theta + 2 \right)$ is

Options

A.-1
B.$4 + \sqrt{10}$
C.$4 - \sqrt{10}$
D.1

Solution

$f(\theta) = \frac{1 + cos2\theta}{2} - 3sin2\theta + 3\left( \frac{1 - cos2\theta}{2} \right) + 2$

$${f(\theta) = 4 - 3sin2\theta - cos2\theta }{f(\theta) \in \lbrack 4 - \sqrt{10},4 + \sqrt{10}\rbrack}$$

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