Question:
If
\[
\tan\theta+\cot\theta=4,
\]
then the value of
\[
\sec^2\theta+\csc^2\theta
\]
is:
If
\[
\tan\theta+\cot\theta=4,
\]
then the value of
\[
\sec^2\theta+\csc^2\theta
\]
is:
Show Hint
A useful identity is
\[
\tan\theta\cot\theta=1.
\]
It greatly simplifies expressions involving both \(\tan\theta\) and \(\cot\theta\).
Updated On: Jun 10, 2026
- \(14\)
- \(16\)
- \(18\)
- \(20\)
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Concept:
The identities
\[
\sec^2\theta=1+\tan^2\theta
\]
and
\[
\csc^2\theta=1+\cot^2\theta
\]
allow us to rewrite the required expression in terms of
\[
\tan\theta+\cot\theta.
\]
Step 1: Square the given relation. \[ (\tan\theta+\cot\theta)^2=16. \] \[ \tan^2\theta+\cot^2\theta+2=16. \] \[ \tan^2\theta+\cot^2\theta=14. \]
Step 2: Use standard identities. \[ \sec^2\theta+\csc^2\theta = (1+\tan^2\theta)+(1+\cot^2\theta). \] \[ = 2+\tan^2\theta+\cot^2\theta. \]
Step 3: Substitute the obtained value. \[ = 2+14. \] \[ =16. \]
Step 4: Final Conclusion. \[ \boxed{16} \] Hence the correct answer is \[ \boxed{\text{Option (B)}}. \]
Step 1: Square the given relation. \[ (\tan\theta+\cot\theta)^2=16. \] \[ \tan^2\theta+\cot^2\theta+2=16. \] \[ \tan^2\theta+\cot^2\theta=14. \]
Step 2: Use standard identities. \[ \sec^2\theta+\csc^2\theta = (1+\tan^2\theta)+(1+\cot^2\theta). \] \[ = 2+\tan^2\theta+\cot^2\theta. \]
Step 3: Substitute the obtained value. \[ = 2+14. \] \[ =16. \]
Step 4: Final Conclusion. \[ \boxed{16} \] Hence the correct answer is \[ \boxed{\text{Option (B)}}. \]
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