Trigonometric Equations
Mixed Trigonometric-Algebraic Equation — Sum of Solutions
nta_pyq_2024_jan
Grade 11
Question:
The sum of the solutions $x\in\mathbb{R}$ of the equation $\dfrac{3\cos2x+\cos^32x}{\cos^6x-\sin^6x}=x^3-x^2+6$ is
Step-by-Step Solution
Key Concept: Simplify LHS: $\frac{\cos2x(3+\cos^22x)}{\cos^6x-\sin^6x}$. Denominator $=\cos^2x-\sin^2x)(\cos^4x+\cos^2x\sin^2x+\sin^4x)=\cos2x(1-\sin^2x\cos^2x)$. Numerator simplification leads to LHS $=4$. So RHS$=x^3-x^2+2=0$ has only $x=-1$ as real root.
LHS $=4$. $x^3-x^2+2=0\Rightarrow x=-1$. Sum of solutions $=-1$.
Correct Answer: 3