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ProveThat, $1 + \frac{\cos 105 + cos165}{\sin 105 + sin375}$ = 0

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ProveThat, $1 + \frac{\cos 105 + cos165}{\sin 105 + sin375}$ = 0

Solution:

$L\ldotp H\ldotp S: 1 + \frac{\cos 105° + cos165°}{\sin 105° + sin375°}$
= $1 + \frac{2cos\frac{105° + 165°}{2}\cos \frac{105° – 165°}{2}}{2sin\frac{105° + 375°}{2}\cos \frac{105° – 375°}{2}}$
= $1 + \frac{2cos\frac{270°}{2}\cos \left( – \frac{60°}{2}\right)}{2sin\frac{480°}{2}\cos \left( – \frac{270°}{2}\right)}$
= $1 + \frac{2cos135°cos30°}{2sin240°cos135°}$
= $1 + \frac{\cos 30°}{\sin 240°}$
=$1 + \frac{\cos 30°}{\sin (180° + 60°)}$
= $1 + \frac{\cos 30°}{ – sin60°}$
= $1 – \frac{\cos 30°}{\sin 60°}$
= $1 – \frac{\cos 30°}{\cos 30°}$
= 1 – 1
= 0 = $R\ldotp H\ldotp S$ [Proved]

আরও দেখুন

  1. ProveThat,$\tan\theta tan\left(\frac{\pi }{3}+\theta \right)\tan \left(\frac{\pi }{3}-\theta \right)$=$tan3\theta$
  2. If sinθ + sinΦ=a and cos θ +cosΦ=b prove that $tan\frac{\theta  – \phi }{2}$ =  $\pm \sqrt{\frac{4 – a^2 – b^2}{a^2 + b^2}}$
  3. Prove That, $sec\left(\frac{\pi }{4} + \theta \right)\sec \left(\frac{\pi }{4} – \theta \right)$ = $2sec2\theta$
  4. Prove that, $cos(\alpha + \beta ) + sin(\alpha – \beta )$ = $2sin\left(\frac{\pi }{4} + \alpha \right)\cos \left(\frac{\pi }{4} + \beta \right)$
  5. if a cosφ=b cosθ then show that atanθ+btanφ=(a+b) tan$\frac{{\mathrm{\theta}} + {\mathrm{\phi}}}{2}$

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