The number of distinct solutions of the equation \frac{5}{4} \left(cos\right)^{2} 2 x + \left(\textrm{ }\text{cos}\right)^{\text{4}} \text{x}\textrm{ } + \left(\textrm{ }\text{sin}\right)^{\text{4}} \text{x}\textrm{ } + \left(\textrm{ }\text{cos}\right)^{\text{6}} \text{x} \textrm{ } + \left(\textrm{ }\text{sin}\right)^{\text{6}} \text{x}\textrm{ } = \textrm{ }\text{2} in the interval [0, 2π] is

Engineering

Mathematics

General Solution of Trigonometric Equation

Question

The number of distinct solutions of the equation $\frac{5}{4} \cos^{2} 2 x + {cos}^{4} x + {sin}^{4} x + {cos}^{6} x + {sin}^{6} x = 2$ in the interval [0, 2π] is

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$\frac{5}{4}$ cos²2x + 1 – 2sin²x cos²x + 1 – 3sin²x cos²x = 2

⇒ 5cos²2x + 8 – 5sin²2x = 8

⇒ cos 4x = 0 ⇒ 4x = (2n + 1) $\frac{π}{2}$ ⇒ x = (2n + 1) $\frac{π}{8}$ , n $\in$ I

$∴ x = \frac{π}{8}, \frac{3 π}{8}, \frac{5 π}{8}, . ............, \frac{15 π}{8} .$

$∴$ Number of solutions is 8.