Question

\(\displaystyle \int \big(\cos\theta\,\csc^{2}\theta-\cos\theta\,\cot^{2}\theta\big)\,d\theta=\ \ ?\)

• A. \(\log\csc\theta+\cot\theta+k\)
• B. \(\csc\theta\cot\theta+k\)
• C. \(k+\sin\theta\)
• D. \(\theta+k\)

Hint:

Use $\csc^2-\cot^2=1$ (parallel to $\sec^2-\tan^2=1$).

Answer 1

The Correct Option is C

Factor \(\cos\theta\): \(\cos\theta(\csc^{2}\theta-\cot^{2}\theta)\). Identity: \(\csc^{2}\theta-\cot^{2}\theta=1\). So integral \(\int \cos\theta\,d\theta=\sin\theta+k\).