Question:
If
\[
\csc\theta+\cot\theta=5
\]
then
\[
11+12\tan\theta=
\]
If
\[
\csc\theta+\cot\theta=5
\]
then
\[
11+12\tan\theta=
\]
Show Hint
For \(\csc\theta \pm \cot\theta\), always use the product identity to simplify quickly.
Updated On: Jul 15, 2026
- \(6\)
- \(9\)
- \(14\)
- \(16\)
Show Solution
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The Correct Option is D
Solution and Explanation
Given:
\[
\csc\theta+\cot\theta=5
\]
Using identity:
\[
(\csc\theta+\cot\theta)(\csc\theta-\cot\theta)=1
\]
So:
\[
\csc\theta-\cot\theta=\frac15
\]
Add:
\[
2\csc\theta=5+\frac15=\frac{26}{5}
\Rightarrow \csc\theta=\frac{13}{5}
\]
Subtract:
\[
2\cot\theta=5-\frac15=\frac{24}{5}
\Rightarrow \cot\theta=\frac{12}{5}
\]
Thus:
\[
\tan\theta=\frac{5}{12}
\]
Now:
\[
11+12\tan\theta=11+12\left(\frac{5}{12}\right)=11+5=16
\]
Hence,
\[
\boxed{16}
\]
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