Another Explanation (5):
Solution
প্রশ্নের সমাধান:
প্রদত্ত:
\(\sin \theta = \frac{4}{5}\) এবং \(\frac{\pi}{2} < \theta < \pi\)
ধাপ 1: \(\cos \theta\) নির্ণয়:
যেহেতু \(\sin^2 \theta + \cos^2 \theta = 1\), তাই:
\[
\cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25}
\]
এবং যেহেতু \(\theta\) দ্বিতীয় কোণের চতুর্দিকে (second quadrant), যেখানে \(\cos \theta < 0\), তাই:
\[
\cos \theta = -\frac{3}{5}
\]
ধাপ 2: \(\tan \theta\) নির্ণয়:
\[
\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{4}{5}}{-\frac{3}{5}} = -\frac{4}{3}
\]
ধাপ 3: \(\sec \theta\) নির্ণয়:
\[
\sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{3}{5}} = -\frac{5}{3}
\]
ধাপ 4: \(\sec(-\theta)\):
\[
\sec(-\theta) = \sec \theta = -\frac{5}{3}
\]
ধাপ 5: \(\cot \theta\):
\[
\cot \theta = \frac{1}{\tan \theta} = \frac{1}{-\frac{4}{3}} = -\frac{3}{4}
\]
ধাপ 6: \(\csc(-\theta)\):
\[
\csc(-\theta) = -\csc \theta
\]
এবং,
\[
\csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{4}{5}} = \frac{5}{4}
\]
অতএব,
\[
\csc(-\theta) = -\frac{5}{4}
\]
ধাপ 7: মূল সমীকরণ:
\[
\frac{\tan \theta + \sec(-\theta)}{\cot \theta + \csc(-\theta)} = \frac{-\frac{4}{3} + (-\frac{5}{3})}{-\frac{3}{4} + (-\frac{5}{4})}
\]
ধাপ 8: সাধারণ করা:
\[
= \frac{-\frac{4}{3} - \frac{5}{3}}{-\frac{3}{4} - \frac{5}{4}} = \frac{-\frac{9}{3}}{-\frac{8}{4}} = \frac{-3}{-2} = \frac{3}{2}
\]
উত্তর:
\(\boxed{\frac{3}{2}}\)
