Question

If $1+\sin^2(2A)=3\sin A\cos A$, then what are the possible values of $\tan A$?

• A. $1/4,\,2$
• B. $1/6,\,3$
• C. $1/2,\,1$
• D. $1/8,\,4$

Hint:

When given a trigonometric identity with $\sin A$ and $\cos A$, divide by $\cos^2 A$ to convert into a quadratic in $\tan A$.

Answer 1

The Correct Option is C

Step 1: Rewrite \(1\) as \(\sin^2A+\cos^2A\): \[ 2\sin^2A-3\sin A\cos A+\cos^2A=0 \]

Step 2: Factor the homogeneous expression: \[ (2\sin A-\cos A)(\sin A-\cos A)=0 \]

Step 3: Each factor gives a value of \(\tan A\): \[ \tan A=\boxed{\dfrac12 \text{ or } 1} \]