Chapter 12

If \(a \cos 2 \theta+b \sin 2 \theta=c\) has \(\alpha\) and \(\beta\) as its solutions, then prove that \(\tan \alpha+\tan \beta=\frac{2 b}{c+a}\) and \(\tan \alpha \tan \beta=\frac{c-a}{c+a} .\)

If \(\alpha\) and \(\beta\) are the solutions of \(a \cos \theta+b \sin \theta=c\), then show that i. \(\cos \alpha+\cos \beta=\frac{2 a c}{a^{2}+b^{2}}\) ii. \(\cos \alpha \cos \beta=\frac{c^{2}-b^{2}}{a^{2}+b^{2}}\) iii. \(\sin \alpha+\sin \beta=\frac{2 b c}{a^{2}+b^{2}}\) iv. \(\sin \alpha \sin \beta=\frac{c^{2}-a^{2}}{a^{2}+b^{2}}\). v. \(\cos (\alpha+\beta)=\frac{a^{2}-b^{2}}{a^{2}+b^{2}}\).

If \(\alpha\) and \(\beta\) are the roots of the equation \(a \sin ^{2} \theta+b \sin \theta+c=0\), show that \(\cos (\alpha+\beta) \cos (\alpha-\beta)=\frac{a^{2}-b^{2}+2 a c}{a^{2}}\)

If \(\alpha\) and \(\beta\) are distinct roots of the equation \(a \cos \theta+b \sin \theta=c\), between 0 and \(2 \pi\), and if \(\alpha+\beta\) also satisfies the equation, show that \(a=c\).

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 \(\pi\).

\begin{aligned} &\text { If } \alpha, \beta, \gamma, \delta \text { are the roots of the equation } \tan \left(\frac{\pi}{4}+\theta\right)=3 \tan 3 \theta \text { , no two which have equal tangents, show }\\\ &\text { that } \tan \alpha+\tan \beta+\tan \gamma+\tan \delta=0 \text { . } \end{aligned}

If \(\theta_{1}\) and \(\theta_{2}\) are two distinct values of \(\theta, 0 \leq \theta_{1}, \theta_{2} \leq 2 \pi\), satisfying the equation \(\sin (\theta+\alpha)=\frac{1}{2} \sin 2 \alpha\) prove that \(\frac{\sin \theta_{1}+\sin \theta_{2}}{\cos \theta_{1}+\cos \theta_{2}}=\cot \alpha\)

Prove that the equation \(x+\frac{1}{x}=\sin \theta\) is not possible for any real value of \(x\).

Find the values of \(\cos \theta\) for which the equation \(2 \cos \theta=x+\frac{1}{x}\) is possible, \(x\) being real

Equation \(2 \sin e^{x}=5^{x}+5^{-x}\) has how many solutions?

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 \(x=y\).

For what values of \(c\), the equation \(\sec \theta+\operatorname{cosec} \theta=c\) has two real roots between 0 and \(2 \pi\) ?

If \(|k|<5\) and \(0^{\circ} \leq \theta \leq 360^{\circ}\), then what is the number of different solutions of \(3 \cos \theta+4 \sin \theta=k\) ?

Equation \(\sin x=\ln x\) has how many solutions?

Equation \(\sin x=\log x\) has how many solutions?

Equation \(x \sin x=1\) has how many solutions?

Equation \(\sec e^{x}=\tanh x^{3}\) has how many solutions?

Equation \(|\ln | x \|=\cot ^{-1} x\) has how many solutions?

Evaluate:i. \(\quad \cos ^{-1} \frac{1}{2}+2 \sin ^{-1} \frac{1}{2} \cdot\left\\{\right. ii. \)2 \sin ^{-1}\left(-\frac{\sqrt {3}}{2}\right)+\cot ^{-1}(-1)+\cos ^{-1}\left(\frac{1}{\sqrt{2}}\right)+\frac{1}{2} \cos ^{-1}(-1)\( \\{Ans. \)\left.\frac{5 \pi}{6}\right\\}\( iii. \)\quad \tan \left(5 \tan ^{-1} \frac{1}{\sqrt{3}}-\frac{1}{4} \sin ^{-1} \frac{\sqrt{3}}{2}\right)\( \\{Ans. -1\\} iv. \)\quad \sin \left(3 \tan ^{-1} \sqrt{3}+2 \cos ^{-1} \frac{1}{2}\right)\( \\{Ans. \)\left.-\frac{\sqrt{3}}{2}\right\\}\( v. \)\cos \left(3 \sin ^{-1} \frac{\sqrt{3}}{2}+\cos ^{-1}\left(-\frac{1}{2}\right)\right)\( \\{ Ans. \)\left.\frac{1}{2}\right\\}\( viii. \)\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)\(. \\{Ans. \)\left.\frac{5 \pi}{6}\right\\}\( ix. \)\quad \cos ^{-1}\left(-\cos \frac{3 \pi}{4}\right)\left\\{\right.\( Ans. \)\left.\frac{\pi}{4}\right\\}\( x. \)\quad \cos ^{-1}(\cos 6) .\\{\( Ans. \)2 \pi-6\\}\( xi. \)\quad \sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)\(. \)\left\\{\right.\( Ans. \)\left.\frac{\pi}{3}\right\\}\( xii. \)\sin ^{-1}\left(-\sin \frac{7 \pi}{3}\right)\( \\{Ans. \)\left.-\frac{\pi}{3}\right\\}\( xiii. \)\sin ^{-1}(\sin 2) .\\{\( Ans. \)\pi-2\\}\( xiv. \)\quad \sin ^{-1}(\sin 10) .\\{\( Ans. \)3 \pi-10\\}\( xv. \)\tan ^{-1}\left(\tan \frac{3 \pi}{10}\right)\( \\{Ans. \)\left.\frac{3 \pi}{10}\right\\}\( xvi. \)\tan ^{-1}\left(-\tan \frac{2 \pi}{3}\right)\( \\{Ans. \)\left.\frac{\pi}{3}\right\\}\( xvii. \)\tan ^{-1}(\tan 5)\( \\{ Ans. \)\left.5-2 \pi\right\\}\( xviii. \)\sin ^{-1}\left(\sin \frac{33 \pi}{7}\right)+\cos ^{-1}\left(\cos \frac{46 \pi}{7}\right)\( \\{ Ans. \)\left.\frac{6 \pi}{7}\right\\}\( xix. \)\quad \tan ^{-1}\left(-\tan \frac{13 \pi}{8}\right)+\cot ^{-1}\left(\cot \left(-\frac{19 \pi}{8}\right)\right)\\{\( Ans. \)\pi\\}\( x. \)\quad \sin ^{-1}\left(\frac{3}{5}\right)+\tan ^{-1}\left(\frac{1}{7}\right)\(. \\{Ans. \)\left.\frac{\pi}{4}\right\\}\( xxi. \)\quad \sin ^{-1}\left(\cos \left(\frac{33 \pi}{5}\right)\right)\(. \\{Ans. - \)\left.\frac{\pi}{10}\right\\}\( xxii. \)\sin \left(\frac{1}{2} \sin ^{-1}\left(-\frac{2 \sqrt{2}}{3}\right)\right)\( \\{Ans. \)\left.-\frac{1}{\sqrt{3}}\right\\}\( xxiii. \)\tan \left(\frac{1}{2} \sin ^{-1}\left(\frac{5}{13}\right)\right)\( \\{ Ans. \)\left.\frac{1}{5}\right\\}\( xxiv. \)\tan \left(\frac{1}{2} \cos ^{-1}\left(\frac{\sqrt{5}}{3}\right)\right) \cdot\left\\{\right.\( Ans. \)\left.\frac{3-\sqrt{5}}{2}\right\\}\( xxv. \)\cot \left(\frac{1}{2} \cos ^{-1}\left(-\frac{4}{7}\right)\right)\( \\{ Ans. \)\left.\sqrt{\frac{3}{11}}\right\\}\( xxvi. \)\tan \left(\cos ^{-1} \frac{4}{5}+\tan ^{-1} \frac{2}{3}\right)\(. \\{Ans. \)\left.\frac{17}{6}\right\\}\( xxvii. \)\sin \left(\tan ^{-1} \frac{8}{15}-\sin ^{-1} \frac{8}{17}\right)\\{\( Ans. 0\)\\}\( vi. \)\quad \cos ^{-1}\left(\cos \frac{\pi}{4}\right)\( \\{ Ans. \)\left.\frac{\pi}{4}\right\\}\( vii. \)\quad \cos ^{-1}\left(\cos \left(\frac{5 \pi}{4}\right)\right) \cdot\left\\{\right.\( Ans. \)\left.\frac{3 \pi}{4}\right\\}$

If \(\sin ^{-1} x=\frac{\pi}{5}\), find the value of \(\cos ^{-1} x\).

If \(\sin ^{-1} x+\sin ^{-1} y=\frac{2 \pi}{3}\), then find the value of \(\cos ^{-1} x+\cos ^{-1} y\).

If \(\frac{\pi}{2} \leq x \leq \frac{3 \pi}{2}\), then find the value of \(\sin ^{-1}(\sin x)\).

If \(\pi \leq x \leq 2 \pi\), then find the value of \(\cos ^{-1}(\cos x)\).

Prove that:- i. \(\quad \sin ^{-1} \frac{3}{5}+\sin ^{-1} \frac{8}{17}=\sin ^{-1} \frac{77}{85}\). ii. \(\quad \sin ^{-1} \frac{4}{5}+\sin ^{-1} \frac{15}{17}=\pi-\sin ^{-1} \frac{77}{85}\). iii. \(\quad \sin ^{-1} \frac{5}{13}+\sin ^{-1} \frac{7}{25}=\cos ^{-1} \frac{253}{325}\). iv. \(\quad \cos ^{-1} \frac{4}{5}+\tan ^{-1} \frac{3}{5}=\tan ^{-1} \frac{27}{11}\). v. \(\quad \cos ^{-1} \frac{4}{5}+\cos ^{-1} \frac{12}{13}=\cos ^{-1} \frac{33}{65}\). vi. \(\quad 2 \cos ^{-1} \frac{3}{\sqrt{13}}+\cot ^{-1} \frac{16}{63}+\frac{1}{2} \cos ^{-1} \frac{7}{25}=\pi\). vii. \(\tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{3}=45^{\circ}\). viii. \(\sin ^{-1} \frac{1}{\sqrt{5}}+\cot ^{-1} 3=45^{\circ}\). ix. \(\quad 2 \tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{7}\right)=\frac{\pi}{4}\). x. \(\quad \tan ^{-1} \frac{1}{7}+\tan ^{-1} \frac{1}{13}=\tan ^{-1} \frac{2}{9}\). xi. \(\tan ^{-1} \frac{2}{3}=\frac{1}{2} \tan ^{-1} \frac{12}{5}\). xii. \(\quad \tan ^{-1} \frac{1}{4}+\tan ^{-1} \frac{2}{9}=\frac{1}{2} \cos ^{-1} \frac{3}{5}\). \begin{aligned} &\text { xiii. } \cos ^{-1}\left(\frac{15}{17}\right)+2 \tan ^{-1}\left(\frac{1}{5}\right)=\tan ^{-1} \frac{171}{140} \\ &\text { xiv. } 2 \tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{7}+2 \tan ^{-1} \frac{1}{8}=\frac{\pi}{4} . \\ &\text { xv. } \tan ^{-1} \frac{3}{4}+\tan ^{-1} \frac{3}{5}-\tan ^{-1} \frac{8}{19}=\frac{\pi}{4} \\ &\text { xvi. } \tan ^{-1} \frac{1}{3}+\tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{7}+\tan ^{-1} \frac{1}{8}=\frac{\pi}{4} \\ &\text { xvii. } 3 \tan ^{-1} \frac{1}{4}+\tan ^{-1} \frac{1}{20}=\frac{\pi}{4}-\tan ^{-1} \frac{1}{1985} . \\ &\text { xviii. } 4 \tan ^{-1} \frac{1}{5}-\tan ^{-1} \frac{1}{70}+\tan ^{-1} \frac{1}{99}=\frac{\pi}{4} . \\ &\text { xix. } \sin ^{-1} \frac{3}{5}+\cos e c^{-1} \frac{5}{4}=\frac{\pi}{2} . \\ &\text { xx. } \tan ^{-1} \frac{120}{119}=2 \sin ^{-1} \frac{5}{13} \text { . } \\ &\text { xxi. } \cos \left(2 \tan ^{-1} \frac{1}{7}\right)=\sin \left(4 \tan ^{-1} \frac{1}{3}\right) . \\ &\text { xxii. } \tan ^{-1} t+\tan ^{-1} \frac{2 t}{1-t^{2}}=\tan ^{-1} \frac{3 t-t^{3}}{1-3 t^{2}}, t \in(-\infty,-1) \cup\left(-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right) \cup(1, \infty) \\ &=\pi+\tan ^{-1} \frac{3 t-t^{3}}{1-3 t^{2}}, t \in\left(\frac{1}{\sqrt{3}}, 1\right) \\ &=-\pi+\tan ^{-1} \frac{3 t-t^{3}}{1-3 t^{2}}, t \in\left(-1,-\frac{1}{\sqrt{3}}\right) \end{aligned}

\(\begin{aligned} \text { Prove that } \cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right) &=2 \tan ^{-1} x, \quad x \geq 0 \\\ &=-2 \tan ^{-1} x, \quad x<0 \end{aligned}\)

\(\tan ^{-1}\left(x^{2}-3 x+3\right)=\frac{\pi}{4}\)

\(\tan ^{-1} 3 x-\cot ^{-1} 3 x=\frac{\pi}{4}\)

\(2\left(\sin ^{-1} x\right)^{2}-5 \sin ^{-1} x+2=0\)

\(4 \tan ^{-1} x-6 \cot ^{-1} x=\pi\)

\(2 \sin ^{-1} x+\cos ^{-1}(1-x)=0\)

\(\sin ^{-1} x+\sin ^{-1}(1-x)=\cos ^{-1} x\)

\(\cos ^{-1} x=\tan ^{-1} x\)

\(\sin ^{-1} x-\cos ^{-1} x=\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)\)

\(\sin ^{-1} \frac{2}{3 \sqrt{x}}-\sin ^{-1} \sqrt{1-x}=\sin ^{-1} \frac{1}{3}\)

\(\tan ^{-1} \frac{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}=\frac{\pi}{4}\)

\(\tan ^{-1} 2 x+\tan ^{-1} 3 x=\frac{\pi}{4}\)

\(\tan ^{-1} \frac{x-1}{x-2}+\tan ^{-1} \frac{x+1}{x+2}=\frac{\pi}{4}\)

$$ \tan ^{-1}(x+1)+\cot ^{-1}(x-1)=\sin ^{-1} \frac{4}{5}+\cos ^{-1} \frac{3}{5} $$

$$ \tan ^{-1}(1+x)+\tan ^{-1}(1-x)=\frac{\pi}{2} $$

$$ \tan ^{-1}(x+1)+\tan ^{-1}(x-1)=\tan ^{-1} \frac{8}{31} $$

$$ 2 \tan ^{-1}(\cos x)=\tan ^{-1}(2 \operatorname{cosec} x) $$

$$ \tan ^{-1} x+2 \cot ^{-1} x=\frac{2}{3} \pi $$

$$ \tan \cos ^{-1} x=\sin \cot ^{-1} \frac{1}{2} $$

$$ \cot ^{-1} x-\cot ^{-1}(x+2)=15^{\circ} $$

$$ \cos ^{-1} \frac{x^{2}-1}{x^{2}+1}+\tan ^{-1} \frac{2 x}{x^{2}-1}=\frac{2 \pi}{3} $$

$$ \cot ^{-1} x+\cot ^{-1}(10-x)=\cot ^{-1} 2 $$

$$ \sin ^{-1} x+\sin ^{-1} 2 x=\frac{\pi}{3} $$

$$ \sin ^{-1} \frac{5}{x}+\sin ^{-1} \frac{12}{x}=\frac{\pi}{2} $$

$$ \sec ^{-1} \frac{x}{4}-\sec ^{-1} \frac{x}{3}=\sec ^{-1} 3-\sec ^{-1} 4 $$

$$ \sin \left(\sin ^{-1} \frac{1}{5}+\cos ^{-1} x\right)=1 $$