Trigonometry & Inverse Trigonometry
Trig Ratios Functions Identities
nta_abhyas_2025
Grade 11
Question:
The value of the expression $\frac{2\cos^4 + \cos^4^3 - \cos^4 \alpha + \cdots - \cos^4}{2\cos^4 + \cos^4^3 - \cos^4 - \cdots - \cos^4 + 1}$ is equal to
$\sqrt{2}$
$\frac{3\sqrt{2}}{4}$
$\frac{1}{2}$
$0$
Step-by-Step Solution
Key Concept: Using double angle formulas and Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ to simplify trigonometric expressions
The expression $2\sin^2\theta + 2(\cos^2\theta - \sin^2\alpha) - 1$ simplifies by first recognizing that $2\sin^2\theta = 1 - \cos(2\theta)$. Expanding the given form and using the identity $2\cos^2\theta - 1 = \cos(2\theta)$, we can rewrite the expression. After careful algebraic manipulation using $\cos^2\alpha + \sin^2\alpha = 1$, the numerator becomes $2[\cos(\alpha)\cos(\alpha) + \cos\theta]$ which simplifies to $\frac{3\cos(2\theta) - \cos(2\theta) - \cos(2\theta)}{1} = 2 - \frac{4}{\sqrt{3}}$. Rationalizing gives $\sqrt{3}$.
Correct Answer: 1