Tan Theta Equation 27Aeec

1. **State the problem:** Solve the equation $$3\tan^2\theta = \frac{2}{\tan^2\theta}$$ for $$0 < \theta < x$$. 2. **Rewrite the equation:** Multiply both sides by $$\tan^2\theta$$ to eliminate the denominator: $$3\tan^2\theta \times \tan^2\theta = \frac{2}{\tan^2\theta} \times \tan^2\theta$$ which simplifies to $$3\tan^4\theta = 2$$ 3. **Isolate $$\tan^4\theta$$:** $$\tan^4\theta = \frac{2}{3}$$ 4. **Take the square root of both sides:** $$\sqrt{\tan^4\theta} = \sqrt{\frac{2}{3}}$$ which gives $$\tan^2\theta = \pm \sqrt{\frac{2}{3}}$$ Since $$\tan^2\theta$$ is always non-negative, discard the negative root: $$\tan^2\theta = \sqrt{\frac{2}{3}}$$ 5. **Take the square root again to solve for $$\tan\theta$$:** $$\tan\theta = \pm \sqrt{\sqrt{\frac{2}{3}}} = \pm \left(\frac{2}{3}\right)^{\frac{1}{4}}$$ 6. **Final answer:** $$\tan\theta = \pm \left(\frac{2}{3}\right)^{\frac{1}{4}}$$ This gives the values of $$\theta$$ where $$\tan\theta$$ equals these values within the domain $$0 < \theta < x$$.