যদি tanθ=1/√7 হয় ,তবে  (cosec^2theta-sec^2theta)/(cosec^2theta+sec^2theta) এর মান কত ?

দেওয়া আছে, \( \tan \theta = \frac{1}{\sqrt{7}} \)

আমরা জানি, \( \sec^2 \theta = 1 + \tan^2 \theta \) এবং \( \cosec^2 \theta = 1 + \cot^2 \theta \)

যেহেতু \( \tan \theta = \frac{1}{\sqrt{7}} \), তাই \( \cot \theta = \frac{1}{\tan \theta} = \sqrt{7} \)

এখন, \( \sec^2 \theta = 1 + \left(\frac{1}{\sqrt{7}}\right)^2 = 1 + \frac{1}{7} = \frac{8}{7} \)

এবং, \( \cosec^2 \theta = 1 + (\sqrt{7})^2 = 1 + 7 = 8 \)

সুতরাং, \( \frac{\cosec^2 \theta - \sec^2 \theta}{\cosec^2 \theta + \sec^2 \theta} = \frac{8 - \frac{8}{7}}{8 + \frac{8}{7}} \)

= \( \frac{\frac{56 - 8}{7}}{\frac{56 + 8}{7}} = \frac{\frac{48}{7}}{\frac{64}{7}} \)

= \( \frac{48}{64} = \frac{3 \times 16}{4 \times 16} = \frac{3}{4} \)

অতএব, \( \frac{\cosec^2 \theta - \sec^2 \theta}{\cosec^2 \theta + \sec^2 \theta} \) এর মান \( \frac{3}{4} \)। 🎉