If $\frac{\sin^{4} x}{2} + \frac{\cos^{4} x}{3} = \frac{1}{5}$, then

If (sin⁴ x)/2 + (cos⁴ x)/3 = 1/5, then $\dfrac{\sin^{4} x}{2} + \dfrac{\cos^{4} x}{3} = \dfrac{1}{5}$ $\text{Find } \tan^{2} x \ \text{and}\ \dfrac{\sin^{8} x}{8} + \dfrac{\cos^{8} x}{27}$ $s = \sin^{2} x, \quad c = \cos^{2} x$ $s + c = 1$ $\dfrac{s^{2}}{2} + \dfrac{c^{2}}{3} = \dfrac{1}{5}$ $c = 1 - s$ $\dfrac{s^{2}}{2} + \dfrac{(1-s)^{2}}{3} = \dfrac{1}{5}$ $15s^{2} + 10(1-s)^{2} = 6$ $15s^{2} + 10 - 20s + 10s^{2} = 6$ $25s^{2} - 20s + 4 = 0$ $(5s - 2)^{2} = 0 \ \Rightarrow \ 5s - 2 = 0 \ \Rightarrow \ s = \dfrac{2}{5}$ $\sin^{2} x = \dfrac{2}{5}, \quad \cos^{2} x = 1 - \dfrac{2}{5} = \dfrac{3}{5}$ $\tan^{2} x = \dfrac{\sin^{2} x}{\cos^{2} x} = \dfrac{2/5}{3/5} = \dfrac{2}{3}$ $\Rightarrow \text{option (1) is true, option (3) is false}$ $\dfrac{\sin^{8} x}{8} + \dfrac{\cos^{8} x}{27} = \dfrac{s^{4}}{8} + \dfrac{c^{4}}{27}$ $s^{4} = \left(\dfrac{2}{5}\right)^{4} = \dfrac{16}{625}, \quad c^{4} = \left(\dfrac{3}{5}\right)^{4} = \dfrac{81}{625}$ $\dfrac{16/625}{8} + \dfrac{81/625}{27} = \dfrac{2}{625} + \dfrac{3}{625} = \dfrac{5}{625} = \dfrac{1}{125}$ $\Rightarrow \text{option (2) is true, option (4) is false}$ $\boxed{\text{Options (1) and (2) are correct}}$ $\tan^{2} x = \dfrac{2}{3}, \quad \dfrac{\sin^{8} x}{8} + \dfrac{\cos^{8} x}{27} = \dfrac{1}{125}$