91Ó°ÊÓ

If \(\alpha\) and \(\beta\) are acute angles and \(\cos 2 \alpha=\frac{3 \cos 2 \beta-1}{3-\cos 2 \beta}\), prove that \(\tan \alpha: \tan \beta=\sqrt{2}: 1\).

If \(\frac{\sin ^{4} \theta}{a}+\frac{\cos ^{4} \theta}{b}=\frac{1}{a+b}\), then \(\frac{\sin ^{8} \theta}{a^{3}}+\frac{\cos ^{8} \theta}{b^{3}}\) (a) \(\frac{1}{a^{3}+b^{3}}\) (b) \(\frac{1}{(a+b)^{3}}\) (c) \(\frac{1}{(a-b)^{3}}\) (d) None.

If \(\tan ^{3}\left(\frac{\alpha}{2}+\frac{\pi}{4}\right)=\tan \left(\frac{\beta}{2}+\frac{\pi}{4}\right)\), then prove that \(\sin \beta=\frac{\left(3+\sin ^{2} \alpha\right) \sin \alpha}{1+3 \sin ^{2} \alpha}\)

The value of \(\tan \left(\frac{\pi}{7}\right) \tan \left(\frac{2 \pi}{7}\right) \tan \left(\frac{3 \pi}{7}\right)\) is (a) 1 (b) \(\frac{1}{\sqrt{7}}\) (c) \(\sqrt{7}\) (d) None.

If \(\sin \beta=\frac{1}{5} \sin (2 \alpha+\beta)\), then prove that \(\tan (\alpha+\beta)=\frac{3}{2} \tan \alpha\)

If \(\alpha\) and \(\beta\) are the solutions of \(\sin ^{2} x+a \sin x+b=0\) as well as that of \(\cos ^{2} x+c \cos x+d=0\), then \(\sin (\alpha+\beta)\) is (a) \(\frac{2 b d}{b^{2}+d^{2}}\) (b) \(\frac{a^{2}+c^{2}}{2 a c}\) (c) \(\frac{b^{2}+d^{2}}{2 b d}\) (d) \(\frac{2 a c}{a^{2}+c^{2}}\)

If \(\sin x+\sin y=3(\cos x-\cos y)\) then prove that \(\sin (3 x)+\sin (3 y)=0\)

Find the value of \(\tan 70^{\circ}-\tan 20^{\circ}\).

If \(\sec (\varphi-\alpha), \sec \varphi, \sec (\varphi+\alpha)\) are in A.P then prove that \(\cos (\varphi)=\sqrt{2} \cos \left(\frac{\alpha}{2}\right)\)

If \(\sec \theta+\tan \theta=1\), then one of the roots of the equation \(a(b-c) x^{2}+b(c-a) x+c(a-b)=0\) is \(\begin{array}{llll}\text { (a) } \tan \theta & \text { (b) } \sec \theta & \text { (c) } \cos \theta & \text { (d) } \sin \theta \text {. }\end{array}\)