Graphs of Trigonometric Functions - Problem Solving

$$\large\tan(x)+\sec(x)=2\cos(x)$$

Find the number of solutions of $x$ in the interval $[0,2\pi]$ that satisfy the equation above.

$$\large \color{#69047E}{\sin^{-1}} \left [ \color{#3D99F6} {\sin} (\color{#20A900}{10}) \right ] = \, \color{#624F41}?$$

For $x\in\mathbb{Z}$, find the probability that

$$2\sin x^{\circ}<1.$$

$$A = \max_{x \in \mathbb{R}} \left( \log_2 3 \right)^{\sin x }, \qquad B = \max_{x \in \mathbb{R}} \left( \log_3 2 \right)^{\sin x }$$

Which is larger, $A$ or $B?$

$$\large {f(x)= \displaystyle \lim_{n \to \infty} \dfrac{x}{(2\sin x)^{2n}+1}}$$

How many values of $x$ are there from $0$ to $\frac{9\pi}{2}$ (both inclusive) such that $f(x)$ is discontinuous at those values of $x$.

$$A = \sin\left[\sin(1)\right] \\ B = \sin\left[\cos(1)\right] \\ C = \cos\left[\sin(1)\right] \\ D =\cos\left[\cos(1)\right]$$

The above are the values of $A,B,C,$ and $D.$ Which of the answer choices is true?

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Clarification: All angles are measured in radians.

Inspiration.

Find the total number of solutions of the equation

$$2x=3\pi(1-\cos x).$$

$$\large \prod_{r = 1}^{12} \sin (rx) = 0$$

What is the number of solutions of $x$ satisfying the equation above in the interval $(0,\pi]?$

How many real numbers $x$ satisfy $\sin x = \frac{x}{100}?$

For every integer $k,$ we define a function $f_k$ by the formula

$$f_k(x)=100x-k\sin x.$$

What is the smallest positive integer value of $k$ such that, for some real $\alpha$, we have $f_k\big(f_k(\alpha)\big)=\alpha,$ but $f_k(\alpha )\neq \alpha?$

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Details and Assumptions: The function is evaluated in radians. There is no degree symbol in the problem.