if sina=1/2 (x+1/x) show that sin 3a +1/2 (x^3+1/x^3)=0 - Maths - Trigonometric Functions

Question

if sina=1/2 (x+1/x) show that sin 3a +1/2 (x^3+1/x^3)=0

Pronay · Accepted Answer

$\sin 3 a = 3 \sin a - 4 \sin^{3} a$

Consider

$\sin 3 a + \frac{1}{2} (x^{3} + \frac{1}{x^{3}}) = 3 \sin a - 4 \sin^{3} a + \frac{1}{2} (x^{3} + \frac{1}{x^{3}}) = 3 \times \frac{1}{2} (x + \frac{1}{x}) - 4 \times {\frac{1}{8} (x + \frac{1}{x})}^{3} + \frac{1}{2} (x + \frac{1}{x}) (x^{2} - 1 + \frac{1}{x^{2}}) = \frac{1}{2} (x + \frac{1}{x}) (3 - {(x + \frac{1}{x})}^{2} + (x^{2} - 1 + \frac{1}{x^{2}})) = \frac{1}{2} (x + \frac{1}{x}) (3 - (x^{2} + 2 + \frac{1}{x^{2}}) + (x^{2} - 1 + \frac{1}{x^{2}})) = \frac{1}{2} (x + \frac{1}{x}) (3 - x^{2} - 2 - \frac{1}{x^{2}} + x^{2} - 1 + \frac{1}{x^{2}}) = 0$