Trigonometric Equations
Trig Equations Inequations
nta_abhyas_2025
Grade 11
Question:
If $f(x) = \frac{\sin x}{\cos x}$, then the number of solution(s) of the equation $f(\sin\theta) + f(\cos\theta) = \tan\theta$, $\theta \in [0, 10\pi]$ is are
Step-by-Step Solution
Key Concept: Substitute $\sin^2\theta = t$ and simplify the function composition, then use periodicity to count solutions.
Given $f(\sin^2\theta) + f(1 - \sin^2\theta)$ with $\sin^2\theta = t$. Then $f(t) + f(1-t) = \frac{2t^2}{t(1-t)} + \frac{2(1-t)^2}{(1-t)t} = \frac{2t + 2(1-t)^2}{t(1-t)} = \frac{20t^2 - 20t + 2}{20t(1-t)} = \tan^2\theta - 1$. When $\tan\theta = 1$, we get $\tan^2\theta - 1 = 0$. Therefore, the number of solutions in the given domain is 20.
Correct Answer: 20