91Ó°ÊÓ

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?