1. Prove that , $\,(i)\, \cot(\theta+15^{\circ})-\tan(\theta-15^{\circ})=\frac{4 \cos {2\theta}}{1+2\sin{2\theta}}$
Sol. $\,(i)\, \cot(\theta+15^{\circ})-\tan(\theta-15^{\circ})\\=\frac{\cos(\theta+15^{\circ})}{\sin(\theta+15^{\circ})}-\frac{\sin(\theta-15^{\circ})}{\cos(\theta-15^{\circ})}\\=\frac{\cos(\theta+15^{\circ})\cos(\theta-15^{\circ})-\sin(\theta+15^{\circ})\sin(\theta-15^{\circ})}{\sin(\theta+15^{\circ})\cos(\theta-15^{\circ})}\\=\frac{\cos(\theta+15^{\circ}+\theta-15^{\circ})}{\frac 12(\sin{2 \theta}+\sin{30^{\circ}})}\\=\frac{4\cos{2\theta}}{1+2\sin{2\theta}}\,\,\,[\text{Using,}\,\,\sin{30^{\circ}}=\frac 12]$
1. Prove that , $\,(ii)\, \,\tan \left(\frac{\pi}{4}+\theta\right)+\tan \left(\frac{\pi}{4}-\theta\right)=2\sec{2 \theta}$
Sol. $\,\, \tan \left(\frac{\pi}{4}+\theta\right)+\tan \left(\frac{\pi}{4}-\theta\right)\\=\frac{\sin \left(\frac{\pi}{4}+\theta\right)}{\cos\left(\frac{\pi}{4}+\theta\right)}+\frac{\sin\left(\frac{\pi}{4}-\theta\right)}{\cos\left(\frac{\pi}{4}-\theta\right)}\\= \frac{\sin\left(\frac{\pi}{4}+\theta\right)\cos\left(\frac{\pi}{4}-\theta\right)+\cos\left(\frac{\pi}{4}+\theta\right)\sin\left(\frac{\pi}{4}-\theta\right)}{\cos \left(\frac{\pi}{4}+\theta\right)\cos\left(\frac{\pi}{4}-\theta\right)}\\=\frac{\sin\left(\frac{\pi}{4}+\theta+\frac{\pi}{4}-\theta\right)}{ \frac 12\left(\cos\frac{\pi}{2}+\cos{2\theta}\right)}\\=2 \sec{2 \theta}$
$(iii)\,\, \cos^3{\theta}\cos{3\theta}+\sin^3\theta\sin{3\theta}=\cos^3{2\theta}$
Sol. $\,\, \cos^3{\theta}\cos{3\theta}+\sin^3\theta\sin{3\theta}\\=\frac{3\cos\theta+\cos{3\theta}}{4}.\cos{3\theta}+\frac{3\sin \theta-\sin{3\theta}}{4}.\sin{3\theta}\\=\frac 14[3(\cos \theta\cos{3\theta}+\sin \theta \sin{3\theta})\\+(\cos^2{3\theta}-\sin^2{3\theta})]\\=\frac 14(3 \cos (3\theta-\theta)+\cos 2.3\theta)\\=\frac 14(3 \cos 2\theta+\cos 2.3\theta)\\=\cos^32\theta$
$(iv)\,\,\cos^3\alpha+\cos^3(120^{\circ}+\alpha)+\cos^3(240^{\circ}+\alpha)=\frac 34 \cos 3\alpha$
Sol. $\,\,\cos^3\alpha+\cos^3(120^{\circ}+\alpha)+\cos^3(240^{\circ}+\alpha)\\=\frac 14 [\cos 3 \alpha+3\cos\alpha+\cos(360^{\circ}+3\alpha)\\+3\cos(120^{\circ}+\alpha)+\cos(720^{\circ}+3\alpha)\\+3\cos(240^{\circ}+\alpha)]\\=\frac 14[3\cos3\alpha+3\cos\alpha+3\{\cos(120^{\circ}+\alpha)\\+\cos(240^{\circ}+\alpha)\}]\\=\frac 14[3\cos 3\alpha+3\cos \alpha+3.2\cos(180^{\circ}+\alpha) \\ \times \cos60^{\circ}]\\=\frac 34\cos 3\alpha$
Additional Notes : $\,4\cos^3(120^{\circ}+\alpha)=\cos [3(120^{\circ}+\alpha)]\\+3 \cos(120^{\circ}+\alpha),\\ 4\cos^3(240^{\circ}+\alpha)=\cos [3(240^{\circ}+\alpha)]\\+3 \cos(240^{\circ}+\alpha), \\ \cos 3 \alpha=\cos(360^{\circ}+3\alpha)=\cos(720^{\circ}+3\alpha)$
Sol. $\,\tan 70^{\circ}-\tan 50^{\circ}+\tan 10^{\circ}\\=\tan (60^{\circ}+10^{\circ})-\tan(60^{\circ}-10^{\circ})+\tan 10^{\circ}\\=\frac{\tan 60^{\circ}+\tan 10^{\circ}}{1-\tan 60^{\circ} \tan 10^{\circ}}-\frac{\tan 60^{\circ}-\tan 10^{\circ}}{1+\tan 60^{\circ}\tan 10^{\circ}}+\tan 10^{\circ}\\=\frac{\sqrt3+\tan 10^{\circ}}{1-\sqrt3\tan 10^{\circ}}-\frac{\sqrt3-\tan 10^{\circ}}{1+\sqrt3\tan 10^{\circ}}+\tan 10^{\circ}\\=\frac{\sqrt3+3\tan 10^{\circ}+\tan 10^{\circ}+\sqrt3 \tan^210^{\circ}-(\sqrt3-3\tan 10^{\circ}\\-\tan 10^{\circ}+\sqrt3 \tan^210^{\circ})}{1-3\tan^210^{\circ}}\\+\tan 10^{\circ}\\=\frac{8\tan 10^{\circ}}{1-3\tan^210^{\circ}}+\tan 10^{\circ}\\=\frac{9 \tan 10^{\circ}-3\tan^3 10^{\circ}}{1-3\tan^210^{\circ}} \\=3\left( \frac{3\tan 10^{\circ}-3\tan^310^{\circ}}{1-3\tan^210^{\circ}}\right)\\=3\tan30^{\circ}\\=\sqrt3$
$\,(vi)\, \tan \theta \tan(\frac{\pi}{3}+\theta)\tan(\frac{\pi}{3}-\theta)=\tan 3\theta$
Sol. $\,\, \tan \theta \tan(\frac{\pi}{3}+\theta)\tan(\frac{\pi}{3}-\theta)\\=\tan \theta . \frac{\sqrt3+\tan \theta}{1-\sqrt3 \tan \theta}. \frac{\sqrt3-\tan \theta}{1+\sqrt3 \tan \theta}\\=\frac{3\tan \theta-\tan^3\theta}{1-3\tan^2\theta}\\=\tan \theta$
$\,(vii)\, \tan \theta+2\tan 2\theta+4 \tan 4\theta+8\cot 8\theta=\cot \theta$
Sol. $\, \tan \theta+2\tan 2\theta+4 \tan 4\theta+8\cot 8\theta\\=\tan \theta+2\tan 2\theta+4 \tan 4\theta+\frac{8(1-\tan^2 4\theta)}{2\tan 4\theta}\\=\tan \theta+2\tan 2\theta+\frac{8\tan^24\theta+8-8\tan^24\theta}{2\tan 2\theta}\\=\tan \theta+2\tan 2\theta+\frac{4}{\tan 2\theta}\\=\tan \theta+2\tan 2\theta +\frac{4(1-\tan^2 2\theta)}{2\tan 2\theta}\\=\tan \theta+\frac{4 \tan^2 2\theta+4-4\tan^2 2\theta}{2\tan 2\theta}\\=\tan \theta+\frac{2}{\tan 2\theta}\\=\tan \theta+\frac{2(1-\tan^2\theta)}{2\tan \theta}\\=\cot \theta$
$\,(viii)\, \cos \frac{\pi}{11}\cos \frac{2\pi}{11}\cos \frac{3\pi}{11}\cos \frac{4\pi}{11}\cos \frac{5\pi}{11} =\frac{1}{32}$
Sol. Let $\,\,\frac{\pi}{11}=\theta \Rightarrow \theta=11 \pi \\ \therefore P=\cos \theta \cos 2\theta \cos 3\theta \cos 4\theta \cos 5\theta \,\,,\text{say}\\ \Rightarrow 2P\sin \theta=(2 \sin \theta\cos \theta) \cos 2 \theta \cos3 \theta \cos 4 \theta\cos 5 \theta \\ \Rightarrow 2P\sin \theta=\sin 2 \theta\cos 2 \theta\cos3 \theta\cos4 \theta\cos5 \theta \\ \Rightarrow 2^2P\sin \theta=2\sin 2 \theta\cos 2 \theta\cos3 \theta\cos4 \theta\cos5 \theta \\ \Rightarrow 2^2P \sin \theta=\sin 4\theta \cos 4\theta \cos 3\theta \cos 5\theta \\ \Rightarrow 2^3P \sin \theta=(2\sin 4\theta \cos 4\theta )\cos 3\theta \cos 5\theta \\ \Rightarrow 2^3 P\sin \theta=\sin 8\theta \cos 3\theta\cos 5\theta \\~~~~~~~~~~~~~~~~~~~~=\sin(11 \theta-3\theta)\cos 3\theta \cos 5\theta\\~~~~~~~~~~~~~~~~~~~~=\sin(\pi-3\theta)\cos 3\theta\cos 5\theta\\~~~~~~~~~~~~~~~~~~~~=\sin 3\theta \cos 3\theta\cos 5\theta \\ \Rightarrow 2^4P\sin\theta=(2\sin 3\theta\cos 3\theta)\cos 5\theta\\ \Rightarrow 2^4P\sin \theta=\sin 6\theta\cos 5\theta \\~~~~~~~~~~~~~~~~~~~~=\sin(11 \theta-5\theta)\cos 5\theta\\~~~~~~~~~~~~~~~~~~~~=\sin(\pi-5\theta)\cos 5\theta\\~~~~~~~~~~~~~~~~~~~~=\sin 5\theta\cos5\theta\\ \Rightarrow2^5P\sin\theta=2\sin 5\theta\cos 5\theta \\ \Rightarrow 2^5P\sin\theta=\sin (2 \times 5\theta)\\ \Rightarrow 2^5P \sin\theta=\sin 10\theta \\ \Rightarrow 2^5P \sin \theta=\sin(11\theta-\theta)\\ \Rightarrow 2^5P\sin\theta=\sin(\pi-\theta)\\ \Rightarrow 2^5P \sin\theta=\sin \theta \\ \Rightarrow P=\frac{1}{2^5} =\frac{1}{32}$
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