If 15^4+10^4 = 6 for some R, then the value of 27^6+8^6 is equal to:

Question

If $15\sin^4\alpha+10\cos^4\alpha = 6$ for some $\alpha\in\mathbb{R}$, then the value of $27\sec^6\alpha+8\csc^6\alpha$ is equal to:

Accepted Answer

$$15\sin^4\alpha+10\cos^4\alpha = 6(\sin^2\alpha+\cos^2\alpha)^2 = 6\sin^4\alpha+6\cos^4\alpha+12\sin^2\alpha\cos^2\alpha$$

$$9\sin^4\alpha+4\cos^4\alpha-12\sin^2\alpha\cos^2\alpha=0 \implies (3\sin^2\alpha-2\cos^2\alpha)^2=0 \implies \tan^2\alpha=\frac23$$

$$27\sec^6\alpha+8\csc^6\alpha = 27(1+\tan^2\alpha)^3+8(1+\cot^2\alpha)^3 = 27\left(\tfrac53\right)^3+8\left(\tfrac52\right)^3 = 125+125 = 250$$

Solutions · 1