Question

if 2sintheta=x+1/x, prove that sin 3theta +1/2(x3+1/x3)​

Answer 1

✪ᴀɴsᴡᴇʀ✪

☆ɢɪᴠᴇɴ☆

• $2sin(\theta) = x + \frac{1}{x}$

☆ᴛᴏ ᴘʀᴏᴠᴇ☆

• $sin(3\theta) + \frac{1}{2}(x^3 + \frac{1}{x^3}) = 0$

☆ᴘʀᴏᴏғ☆

$\:\:\:\:sin(3\theta)$

$= 3sin(\theta) -4sin^3(\theta)$

$= \frac{3}{2}(x + \frac{1}{x}) -4[\frac{1}{2}(x + \frac{1}{x})]^3$

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

$= \frac{3}{2}(x + \frac{1}{x})$

$\:\:\:\: -\frac{1}{2}[x^3 + \frac{1}{x^3} + 3x.\frac{1}{x}(x + \frac{1}{x})]$

$= \frac{3}{2}(x + \frac{1}{x})-\frac{1}{2}(x^3 + \frac{1}{x^3}) -\frac{3}{2}(x + \frac{1}{x})$

$\implies sin(3\theta) = -\frac{1}{2}(x^3 + \frac{1}{x^3})$

$\implies sin(3\theta) + \frac{1}{2}(x^3 + \frac{1}{x^3}) = 0$

Hence Proved