Write in the simplest form: tan⁻¹((3a²x − x³) / (a³ − 3ax²)), a 0; −a/√3 ≤ x ≤ a/√3

**OR**

Prove that: sin⁻¹(8/17) + sin⁻¹(3/5) = tan⁻¹(77/36).

Question

Write in the simplest form: tan⁻¹((3a²x − x³) / (a³ − 3ax²)), a > 0; −a/√3 ≤ x ≤ a/√3

**OR**

Prove that: sin⁻¹(8/17) + sin⁻¹(3/5) = tan⁻¹(77/36).

Accepted Answer

>

> [!TLDR]
> Substituting $x=a	an	heta$ turns the given expression into the triple-angle tangent formula $	an 3	heta$, so the whole thing collapses to $3	an^{-1}(x/a)$.
>
> Let $x = a	an	heta$, so $	heta = 	an^{-1}(x/a)$.
>
> **Numerator:** $3a^2x - x^3 = 3a^2(a	an	heta) - (a	an	heta)^3 = a^3(3	an	heta - 	an^3	heta)$
>
> **Denominator:** $a^3 - 3ax^2 = a^3 - 3a(a	an	heta)^2 = a^3(1-3	an^2	heta)$
>
> $$\frac{3a^2x-x^3}{a^3-3ax^2} = \frac{3	an	heta-	an^3	heta}{1-3	an^2	heta} = 	an(3	heta)$$
>
> using the standard triple-angle identity $	an3	heta = \dfrac{3	an	heta-	an^3	heta}{1-3	an^2	heta}$.
> …

Q.Write in the simplest form: tan⁻¹((3a²x − x³) / (a³ − 3ax²)), a > 0; −a/√3 ≤ x ≤ a/√3 OR Prove that: sin⁻¹(8/17) + sin⁻¹(3/5) = tan⁻¹(77/36).

Concept understanding — Triple Angle Identity

Triple Angle Identities: From Intuition to Formula

The Precise Statements

Deriving $\sin 3\theta$

Deriving $\cos 3\theta$

Deriving $\tan 3\theta$

What to Remember for Exams …

Q.Write in the simplest form: tan⁻¹((3a²x − x³) / (a³ − 3ax²)), a > 0; −a/√3 ≤ x ≤ a/√3 OR Prove that: sin⁻¹(8/17) + sin⁻¹(3/5) = tan⁻¹(77/36).

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