Question

If \[ \tan\theta+\cot\theta=4, \] then the value of \[ \sec^2\theta+\csc^2\theta \] is:

• A. \(14\)
• B. \(16\)
• C. \(18\)
• D. \(20\)

Hint:

A useful identity is \[ \tan\theta\cot\theta=1. \] It greatly simplifies expressions involving both \(\tan\theta\) and \(\cot\theta\).

Answer 1

The Correct Option is C

Step 1: Note the given fact.
We are told $\tan\theta+\cot\theta=4$, and we must find $\sec^2\theta+\csc^2\theta$.

Step 2: Use a hidden identity.
A useful fact is that $\tan\theta\cdot\cot\theta=1$, because $\cot\theta$ is just $\tfrac{1}{\tan\theta}$. Keep this in mind.

Step 3: Connect to the target.
Recall the two basic identities $\sec^2\theta=1+\tan^2\theta$ and $\csc^2\theta=1+\cot^2\theta$. Adding them, \[ \sec^2\theta+\csc^2\theta=2+\tan^2\theta+\cot^2\theta. \]
Step 4: Square the given relation.
Square $\tan\theta+\cot\theta=4$: \[ \tan^2\theta+2\tan\theta\cot\theta+\cot^2\theta=16. \]
Step 5: Use the product equals one.
Since $\tan\theta\cot\theta=1$, the middle term is $2$. So $\tan^2\theta+\cot^2\theta=16-2=14$.

Step 6: Put it together.
Therefore \[ \sec^2\theta+\csc^2\theta=2+14=18. \]
\[ \boxed{18} \]