Trigonometric Ratios and Identities
Identities
Premium Question
Grade 11
Question:
If $\frac{\sin^4 x}{a} + \frac{\cos^4 x}{b} = \frac{1}{a + b}$, then $\frac{\sin^8 x}{a^3} + \frac{\cos^8 x}{b^3}$ equals:
(1) $\frac{1}{(a + b)^3}$
(2) $\frac{1}{(a + b)^2}$
(3) $\frac{a^3 + b^3}{(a + b)^4}$
(4) $\frac{1}{a^2 + b^2}$
Step-by-Step Solution
Key Concept: The given condition forces $\sin^2 x = a/(a + b)$ and $\cos^2 x = b/(a + b)$. Substitute these into $\sin^8 x / a^3 + \cos^8 x / b^3$ and simplify.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)