মান নির্ণয়

tan⁡(π6+θ)tan⁡(π6−θ)=?\tan \left(\frac{\pi}{6}+\theta\right) \tan \left(\frac{\pi}{6}-\theta\right)= ?

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Solve:tan⁡(π6+θ)tan⁡(π6−θ)=sin⁡(π6+θ)sin⁡(π6−θ)cos⁡(π6+θ)cos⁡(π6−θ)=2sin⁡(π6+θ)sin⁡(π6−θ)2cos⁡(π6+θ)cos⁡(π6−θ)=cos⁡(π6+θ−π6+θ)−cos⁡(π6+θ+π6−θ)cos⁡(π6+θ−π6+θ)+cos⁡(π6+θ+π6−θ)=cos⁡2θ−cos⁡π3cos⁡2θ+cos⁡π3=cos⁡2θ−12cos⁡2θ+12=2cos⁡2θ−12cos⁡2θ+1 (ans ) \begin{array}{l} \tan \left(\frac{\pi}{6}+\theta\right) \tan \left(\frac{\pi}{6}-\theta\right) \\ = \frac{\sin \left(\frac{\pi}{6}+\theta\right) \sin \left(\frac{\pi}{6}-\theta\right)}{\cos \left(\frac{\pi}{6}+\theta\right) \cos \left(\frac{\pi}{6}-\theta\right)} \\ = \frac{2 \sin \left(\frac{\pi}{6}+\theta\right) \sin \left(\frac{\pi}{6}-\theta\right)}{2 \cos \left(\frac{\pi}{6}+\theta\right) \cos \left(\frac{\pi}{6}-\theta\right)} \\ = \frac{\cos \left(\frac{\pi}{6}+\theta-\frac{\pi}{6}+\theta\right)-\cos \left(\frac{\pi}{6}+\theta+\frac{\pi}{6}-\theta\right)}{\cos \left(\frac{\pi}{6}+\theta-\frac{\pi}{6}+\theta\right)+\cos \left(\frac{\pi}{6}+\theta+\frac{\pi}{6}-\theta\right)} \\ = \frac{\cos 2 \theta-\cos \frac{\pi}{3}}{\cos 2 \theta+\cos \frac{\pi}{3}}=\frac{\cos 2 \theta-\frac{1}{2}}{\cos 2 \theta+\frac{1}{2}} \\ = \frac{2 \cos 2 \theta-1}{2 \cos 2 \theta+1}\text { }(\text {ans }) \end{array}

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