Question 8:

The graph of $y = \tan(\cos(\sin(x)))$ has a period of $P$ and a maximum value of $M$. Which row is correct?

Solution:

We analyze the inner structure step by step:

• $\sin(x)$ has period $2\pi$.
• $\cos(\sin(x))$ is also periodic with period $2\pi$, since cosine is even and $\cos(\sin(x)) = \cos(\sin(x+2\pi))$.
• Thus, $\tan(\cos(\sin(x)))$ has fundamental period $P = \pi$, because tangent has period $\pi$.

Next, consider the range: $\sin(x) \in [-1,1] \implies \cos(\sin(x)) \in [\cos(1),1]$. Thus, the argument of tangent lies between $\cos(1)$ and $1$.

So the maximum value: $$M = \tan(1) \approx 1.557 > 1.$$

Therefore: $$P = \pi, \quad M > 1.$$

Answer: (F) $P = \pi, \; M > 1$