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

We are given \[ 1+\sin^2A=3\sin A\cos A. \] Divide both sides by $\cos^2 A$ (valid for $\cos A\neq0$): \[ \frac{1}{\cos^2A}+\tan^2A=3\tan A. \] But $\frac{1}{\cos^2A}=1+\tan^2A$. So \[ 1+\tan^2A+\tan^2A=3\tan A \quad\Rightarrow\quad 1+2\tan^2A=3\tan A. \] Hence quadratic: \[ 2\tan^2A-3\tan A+1=0. \] Solve: \[ \tan A=\frac{3\pm \sqrt{9-8}}{4}=\frac{3\pm1}{4}. \] So $\tan A=\tfrac12$ or $1$. \[ \boxed{\tfrac12,\,1} \]