Chapter 12

$$ \cos 2 x \cos 5 x<\cos 3 x $$

$$ \sin 2 x \sin 3 x-\cos 2 x \cos 3 x>\sin 10 x $$

$$ \cot x+\cot \left(x+\frac{\pi}{2}\right)+2 \cot \left(x+\frac{\pi}{3}\right)>0 $$

$$ 2 \sin ^{2} x-\sin x+\sin 3 x<1 $$

$$ 4 \sin x \sin 2 x \sin 3 x>\sin 4 x $$

$$ \frac{\cos ^{2} 2 x}{\cos ^{2} x} \geq 3 \tan x $$

$$ \frac{\cos x+2 \cos ^{2} x+\cos 3 x}{\cos x+2 \cos ^{2} x-1}>1 $$

$$ \sin ^{-1} x \leq 5 $$

$$ \sin ^{-1} x \geq-2 $$

$$ \cos ^{-1} x \leq \cos ^{-1} \frac{1}{4} $$

$$ \cos ^{-1} x>\frac{\pi}{6} $$

$$ \tan ^{-1} x>-\frac{\pi}{3} $$

$$ \cot ^{-1} x>2 $$

$$ \left(\cot ^{-1} x\right)^{2}-5 \cot ^{-1} x+6>0 $$

$$ \left(\tan ^{-1} x\right)^{2}-4 \tan ^{-1} x+3>0 $$

$$ \log _{2}\left(\tan ^{-1} x\right)>1 $$

$$ 2^{\tan ^{-1} x}+2^{-\tan ^{-1} x} \geq 2 $$

$$ 4\left(\cos ^{-1} x\right)^{2}-1 \geq 0 $$

$$ \sin ^{-1} x<\cos ^{-1} x $$

$$ \cos ^{-1} x>\cos ^{-1} x^{2} $$

$$ \tan ^{-1} x>\cot ^{-1} x $$

$$ \sin ^{-1} x<\sin ^{-1}(1-x) $$

$$ \tan ^{2}\left(\sin ^{-1} x\right)>1 $$

$$ \lim _{x \rightarrow \infty} x\left(\tan ^{-1} \frac{x+1}{x+2}-\frac{\pi}{4}\right) $$

$$ \lim _{x \rightarrow \infty} x\left(\tan ^{-1} \frac{x+1}{x+2}-\tan ^{-1} \frac{x}{x+2}\right) $$

$$ \lim _{x \rightarrow 0} \frac{\sin ^{-1} x-\tan ^{-1} x}{x^{3}} $$

$$ \lim _{x \rightarrow \infty}(x+2) \tan ^{-1}(x+2)-x \tan ^{-1} x $$

Prove that \(\sin \theta \sec 3 \theta=\frac{1}{2}(\tan 3 \theta-\tan \theta)\) and hence find the sum to \(n\) terms of the series \(\sin \theta \sec 3 \theta+\sin 3 \theta \sec 3^{2} \theta+\sin 3^{2} \theta \sec 3^{3} \theta+\ldots \ldots \ldots \ldots \ldots .\)

Prove that \(\tan \alpha=\cot \alpha-2 \cot 2 \alpha .\) Hence show that the sum to \(n\) terms of the series \(\tan \alpha+2 \tan 2 \alpha+2^{2} \tan 2^{2} \alpha+\ldots \ldots \ldots .\) is \(\cot \alpha-2^{n} \cot 2^{n} \alpha\)

If \(a_{1}, a_{2}, a_{3}, \ldots \ldots a_{n}\) are in AP with common difference \(d\), then prove that the sum of the series \(\sin d\left[\sec a_{1} \sec a_{2}+\sec a_{2} \sec a_{3}+\ldots+\sec a_{n-1} \sec a_{n}\right]\), is \(\tan a_{n}-\tan a_{1}\)

If \(a_{1}, a_{2}, a_{3}, \ldots \ldots a_{n}\) are in AP with common difference \(d\), then prove that the sum of the series \(\sin d\left[\operatorname{cosec} a_{1} \operatorname{cosec} a_{2}+\operatorname{cosec} a_{2} \operatorname{cosec} a_{3}+\ldots+\operatorname{cosec} a_{n-1} \operatorname{cosec} a_{n}\right]=\cot a_{1}-\cot a_{n}\)

If \(U_{n}=\sin n \theta \sec ^{n} \theta, V_{n}=\cos n \theta \sec ^{n} \theta, n=0,1,2, \ldots \ldots \ldots\), prove that \(V_{n}-V_{n-1}=-U_{n-1} \tan \theta .\) Hence deduce that \(U_{1}+U_{2}+\ldots \ldots .+U_{n}=\cot \theta \sec ^{n+1} \theta\left(\cos ^{n+1} \theta-\cos (n+1) \theta\right)\).

If \(A+B+C=\pi(A, B, C>0)\) and the angle \(C\) is obtuse, then show that \(\tan A \tan B<1\).

If \(y=\sec ^{-1}\left(\frac{x+1}{x-1}\right)+\sin ^{-1}\left(\frac{x-1}{x+1}\right)\) then find \(\frac{d y}{d x}\). \\{Ans. 0\(\\}\)

Show that \(\sin ^{p} \theta \cos ^{q} \theta\) attains a maxima when \(\theta=\tan ^{-1} \sqrt{\frac{p}{q}}\).

Find the values of \(x\) for which the function \(f(x)=1+2 \sin x+3 \cos ^{2} x, \quad 0 \leq x \leq \frac{2 \pi}{3}\) has maxima or minima. Also find the values of the function at these extremum. \\{Ans. minima at \(x=\frac{\pi}{2}, f\left(\frac{\pi}{2}\right)=3\), maxima at \(\left.x=\sin ^{-1} \frac{1}{3}, f\left(\sin ^{-1} \frac{1}{3}\right)=\frac{13}{3}\right\\}\)

If \(\frac{1}{6} \sin x, \cos x, \tan x\) are in \(\mathrm{GP}\), then find the value of \(x\).

If \(\tan p \theta-\tan q \theta=0\), then show that the values of \(\theta\) form a series in A.P.

For what value of \(b\), will the roots of the equation \(\cos x=b,-1 \leq b \leq 1\) when arranged in ascending order of their magnitudes, form an AP.

Find the greatest and least values of \(\left(\sin ^{-1} x\right)^{3}+\left(\cos ^{-1} x\right)^{3}\).

What is the number of all possible triplets \(\left(a_{1}, a_{2}, a_{3}\right)\) such that \(a_{1}+a_{2} \cos 2 x+a_{3} \sin ^{2} x=0\) for all \(x\) ?

If \(\tan ^{-1} x+\tan ^{-1} y=\frac{\pi}{4}-\tan ^{-1} z\) and \(x+y+z=1\) then arithmetic mean of odd powers of \(x, y, z\) is \(\frac{1}{3}\). (True/False)

Find the intervals of monotonicity of the function \(f(x)=3 \cos ^{4} x+10 \cos ^{3} x+6 \cos ^{2} x-3,0 \leq x \leq \pi\).

Consider the system of linear equations in \(x, y, z\) \((\sin 3 \theta) x-y+z=0\) \((\cos 2 \theta) x+4 y+3 z=0\) \(2 x+7 y+7 z=0\)

Let \(\lambda\) and \(\alpha\) be real. Find the set of all values of \(\lambda\) for which the system of linear equations \(\lambda x+(\sin \alpha) y+(\cos \alpha) z=0\) \(x+(\cos \alpha) y+(\sin \alpha) z=0\) \(-x+(\sin \alpha) y-(\cos \alpha) z=0\) has a non-trivial solution. For \(\lambda=1\), find all values of \(\alpha\).