91Ó°ÊÓ

Express \(6 \cos ^{2} \alpha+8 \sin \alpha \cos \alpha\) as \(A+B \cos (2 \alpha-\beta)\) and hence show that the greatest and the least values of the expression are 8 and \(-2\) respectively.

Prove that \(-4 \leq \cos 2 x+3 \sin x \leq \frac{17}{8}\)

\(\sin 2 x+\cos 2 x=\sin x+\cos x\)

$$ \cot x+\frac{\sin x}{1+\cos x}=2 $$

$$ \sin x \cos x-6 \sin x+6 \cos x+6=0 $$

$$ 2(1-\sin x-\cos x)+\tan x+\cot x=0 $$

$$ \sin ^{10} x+\cos ^{10} x=\frac{29}{16} \cos ^{4} 2 x $$

$$ \sqrt{-3 \sin 5 x-\cos ^{2} x-3}+\sin x=1 $$

If \(\theta_{1}, \theta_{2}, \theta_{3}\) are the values of \(\theta\) which satisfy the equation \(\tan 2 \theta=\lambda \tan (\theta+\alpha)\), and if no two of these values differ by a multiple of \(\pi\), then show that \(\theta_{1}+\theta_{2}+\theta_{3}+\alpha\) is a multiple of \(\pi\).

Prove that the equation \(\sec ^{2} \theta=\frac{4 x y}{(x+y)^{2}}\) is possible for real values of \(x\) and \(y\) only if \(x=y\).