Problem 39 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity. $$\sin ^{1 / 2} x \cos x-\sin ^{5 / 2} x \cos x=\cos ^{3} x \sqrt{\sin x}$$

Step-by-Step Solution

Answer

The given identity \(\sin ^{1 / 2} x \cos x - \sin ^{5 / 2} x \cos x = \cos ^{3} x \sqrt{\sin x}\) holds if \(\cos x = 1\). This will be the case when \(x=2n\pi\), \(n\) being an integer.

Step 1: Rewrite the Equation

Rewrite the equation in a format in terms of sine. Remembering that \(\cos^{2} x = 1 - \sin^{2} x\), this allows the right-hand side (RHS) to be rewritten in terms of \(\sin x\) as, \(\cos^{3}x \sqrt{\sin x}=(1-\sin^{2} x)\sqrt{\sin x}\). Using the distributive property to expand yields, \( \sqrt{\sin x}-\sin^{2} x \sqrt{\sin x}\).

Step 2: Simplify the Right Hand Equation

\(\sqrt{\sin x} - \sin^{2} x \sqrt{\sin x}\) can be simplified to \(\sin ^{1 / 2} x - \sin ^{2.5} x\). The expression \(\sin ^{2.5} x\) is the same as \(\sin^{5 / 2} x\). So this expression simplifies to \(\sin^{1 / 2} x - \sin^{5 / 2} x\).

Step 3: Match the Left and Right Sides

Comparing this with the left-hand side (LHS), \(\sin^{1 / 2} x \cos x - \sin^{5 / 2} x \cos x\), notice that both expressions match if we consider \(\cos x = 1\).