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Magazine Problem 4
Prove that \(\frac{\sin A+\sin 3 A+\sin 5 A+\sin 7 A}{\cos A+\cos 3 A+\cos 5 A+\cos 7 A}\) \(=\tan 4 A\)
Problem 4
Prove that \(\frac{\cos 8 A \cos 5 A-\cos 12 A \cos 9 A}{\sin 8 A \cos 5 A+\cos 12 A \sin 9 A}\) \(=\tan 4.4\)
Problem 5
\(\begin{aligned}&\text { Prove that } & \frac{\sin (A-C)+2 \sin A+\sin (A+C)}{\sin (B-C)+2 \sin B+\sin (B+C)} \\\& & =\frac{\sin A}{\sin B} .\end{aligned}\)
Problem 5
Prove that \(\sin 10^{\circ}+\sin 20^{\circ}+\sin 40^{\circ}+\sin 50^{\circ}=\sin 70^{\circ}\) \(+\sin 80^{\circ}\)
Problem 6
If \(\cos (\alpha-\beta)=1\) and \(\cos (\alpha+\beta)=\frac{1}{e},-\pi<\) \(\alpha, \beta<\pi\), then total number of ordered pair of \((\alpha, \beta)\) is [IIT Screening-2005] (a) 0 (b) 1 (c) 2 (d) 4
Problem 6
\(\begin{aligned}&\text { Prove that } & \frac{\sin 11 A \sin A+\sin 7 A \sin 3 A}{\cos 11 A \sin A+\cos 7 A \sin 3 A} \\\& & =\tan 8 A\end{aligned}\)
Problem 6
Prove that, \(\cos \left(\frac{3 \pi}{4}+x\right)-\cos \left(\frac{3 \pi}{4}-x\right)\) \(=-\sqrt{2} \sin x\)
Problem 7
Prove that \(\sin ^{2}\left(\frac{\pi}{8}+\frac{A}{2}\right)-\sin ^{2}\left(\frac{\pi}{8}-\frac{A}{2}\right)\) $$ =\frac{1}{\sqrt{2}} \sin A $$
Problem 7
Prove that \(\cos A+\cos \left(120^{\circ}-A\right)+\) \(\cos \left(120^{\circ}+A\right)=0\)
Problem 7
If \(\theta, \phi\) are acute, \(\sin \theta=1 / 2, \cos \phi=1 / 3\) then \((\theta+\phi) \in\) [IIT Screening-2004] (a) \((\pi / 3, \pi / 2)\) (b) \((\pi / 2,2 \pi / 3)\) (c) \((2 \pi / 3,5 \pi / 6)\) (d) \((5 \pi / 6, \pi)\)
