Differentiation (Part-38) | S N De

 

$\,1(i).\,$ Prove that $\,\,\sin\alpha+\sin(120^{\circ}+\alpha)+\sin(240^{\circ}+\alpha)=0$

Sol. $\,\,\sin\alpha+\sin(120^{\circ}+\alpha)+\sin(240^{\circ}+\alpha)\\=\sin\alpha+2\sin\left(\frac{120^{\circ}+\alpha+240^{\circ}+\alpha}{2}\right)\\ \times\cos\left(\frac{240^{\circ}+\alpha-120^{\circ}-\alpha}{2}\right)\\=\sin\alpha+2\sin(180^{\circ}+\alpha)\cos60^{\circ}\\=\sin\alpha+2.(-\sin\alpha).\frac 12\\=\sin\alpha-\sin\alpha\\=0$

$\,1(ii).\,$ Prove that $\,\sin5x\cos2x+\cos6x\sin3x=\sin8x\cos x$

Sol. $\,\sin5x\cos2x+\cos6x\sin3x\\=\frac 12\left(2\sin5x\cos2x+2\cos6x\sin3x\right)\\=\frac 12\left[\sin(5x+2x)+\sin(5x-2x)\\+\sin(6x+3x)-\sin(6x-3x)\right]\\=\frac 12(\sin7x+\sin3x+\sin9x-\sin3x)\\=\frac 12(\sin 7x+\sin9x)\\=\frac 12 \times 2\sin\frac{7x+9x}{2}\cos\frac{9x-7x}{2}\\=\sin8x\cos x\,\,\text{(proved)}$

$\,1(iii).\,$ Prove that $\,\,\sec(\pi/4+\theta)\sec(\pi/4-\theta)=2\sec 2\theta$

Sol. $\,\,\sec(\pi/4+\theta)\sec(\pi/4-\theta)\\=\frac{2}{2\cos(\pi/4+\theta)\cos(\pi/4-\theta)}\\=\frac{2}{\cos\left(\frac{\pi}{4}+\theta+\frac{\pi}{4}-\theta\right)+\cos\left(\frac{\pi}{4}+\theta-\frac{\pi}{4}+\theta\right)}\\=\frac{2}{\cos\frac{\pi}{2}+\cos2\theta}\\=\frac{2}{\cos2\theta}\\=2\sec2\theta\,\,\,\text{(proved)}$

$\,1(iv).\,$ Prove that $\,\,\cos(\alpha+\beta)+\sin(\alpha-\beta)=2\sin\left(\frac{\pi}{4}+\alpha\right) \\ \times\cos\left(\frac{\pi}{4}+\beta\right)$

Sol. $\quad2\sin\left(\frac{\pi}{4}+\alpha\right)\cos\left(\frac{\pi}{4}+\beta\right)\\=\sin\left(\frac{\pi}{4}+\alpha+\frac{\pi}{4}+\beta\right)+\sin\left(\frac{\pi}{4}+\alpha-\frac{\pi}{4}-\beta\right)\\=\sin\left[\frac{\pi}{2}+(\alpha+\beta)\right]+\sin(\alpha-\beta)\\=\cos(\alpha+\beta)+\sin(\alpha-\beta)\,\,\text{(proved)}$

$\,1(v).\,$ Prove that $\,\,1+\frac{\cos105^{\circ}+\cos165^{\circ}}{\sin105^{\circ}+\sin375^{\circ}}=0$

Sol. $\,\,1+\frac{\cos105^{\circ}+\cos165^{\circ}}{\sin105^{\circ}+\sin375^{\circ}}\\=1+\frac{2\cos\frac{165^{\circ}+105^{\circ}}{2}\cos\frac{165^{\circ}-105^{\circ}}{2}}{2\sin\frac{375^{\circ}+105^{\circ}}{2}\cos\frac{375^{\circ}-105^{\circ}}{2}}\\=1+\frac{2\cos135^{\circ}\cos30^{\circ}}{2\sin240^{\circ}\cos135^{\circ}}\\=1+\frac{\cos30^{\circ}}{\sin(3\times 90^{\circ}-30^{\circ})}\\=1+\frac{\cos30^{\circ}}{-\cos30^{\circ}}\\=1-1\\=0\,\,\text{(proved)}$

$\,2(i).\,$ Prove that $\,\,\sin10^{\circ}+\sin50^{\circ}-\sin70^{\circ}=0$

Sol. $\,\,\sin10^{\circ}+\sin50^{\circ}-\sin70^{\circ}\\=2\sin\frac{10^{\circ}+50^{\circ}}{2}\cos\frac{50^{\circ}-10^{\circ}}{2}-\sin(90^{\circ}-20^{\circ})\\=2\sin30^{\circ}\cos20^{\circ}-\cos20^{\circ}\\=2.\frac 12.\cos20^{\circ}-\cos20^{\circ}\\=\cos20^{\circ}-\cos20^{\circ}\\=0$

$\,2(ii).\,$ Prove that $\,\,\cos80^{\circ}-\cos40^{\circ}+\sqrt3\cos70^{\circ}=0$

Sol.$\,\,\cos80^{\circ}-\cos40^{\circ}+\sqrt3\cos70^{\circ}\\=2\sin\frac{80^{\circ}+40^{\circ}}{2}\sin\frac{40^{\circ}-80^{\circ}}{2}+\sqrt3\cos(90^{\circ}-20^{\circ})\\=2\sin60^{\circ}\sin(-20^{\circ})+\sqrt3\sin20^{\circ}\\=-2.\frac{\sqrt3}{2}\sin20^{\circ}+\sqrt3\sin20^{\circ}\\=0\,\,\text{(proved)}$

$\,2(iii).\,$ Prove that $\,\,\cos(60^{\circ}+A)+\cos(60^{\circ}-A)-\cos A=0$

Sol. $\,\,\cos(60^{\circ}+A)+\cos(60^{\circ}-A)-\cos A\\=2\cos\frac{60^{\circ}+A+60^{\circ}-A}{2}\cos\frac{60^{\circ}+A-60^{\circ}+A}{2}-\cos A\\=2\cos60^{\circ}\cos A-\cos A\\=2.\frac 12.\cos A-\cos A\\=\cos A-\cos A\\=0\,\,\text{(proved)}$

$\,2(iv).\,$ Prove that $\,\,\sin(\frac{2\pi}{3}+\theta)-\sin(\frac{2\pi}{3}-\theta)+\sin\theta=0$

Sol. $\,\,\sin(\frac{2\pi}{3}+\theta)-\sin(\frac{2\pi}{3}-\theta)+\sin\theta\\=2\sin\left(\frac{\frac{2\pi}{3}+\theta-\frac{2\pi}{3}+\theta}{2}\right)\cos\left(\frac{\frac{2\pi}{3}+\theta+\frac{2\pi}{3}-\theta}{2}\right)+\sin\theta\\=2\sin\theta\cos\frac{2\pi}{3}+\sin\theta\\=2\sin\theta\cos\left(\pi-\frac{\pi}{3}\right)+\sin\theta\\=-2\sin\theta.\cos\frac{\pi}{3}+\sin\theta\\=-2\sin\theta.\frac 12+\sin\theta\\=-\sin\theta+\sin\theta\\=0\,\,\text{(proved)}$

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