Question

Let $f\left(x\right)=\frac{\sin (x-a)+\sin (x+a)}{\cos (x-a)-\cos (x+a)}$, then

• A. $f(x+2\pi )=f\left(x\right)$ but $f(x+\alpha )\ne f\left(x\right)$ for any $0<\alpha <2\pi$
• B. $f$ is strictly increasing function
• C. $f$ is a strictly decreasing function
• D. $f$ is a constant function

Answer 1

Answer: (D)

$f$ is a constant function

$f\left(x\right)=\frac{\sin (x-a)+\sin (x+a)}{\cos (x-a)-\cos (x+a)}$

$\therefore f\left(x\right)=\frac{\sin x\cos a-\cos x\sin a+\sin x\cos a+\cos x\sin a}{\cos x\cos a+\sin x\sin a-\cos x\cos a+\sin x\sin a}$

$=\frac{2\sin x\cos a}{2\sin x\sin a}$

$=\cot a$

$f\left(x\right)$ is thus a constant function since it does not depend on $x$.