Counting solutions

Solve the first problem 16 problems

Number of solutions of a trig-root equation ●●○○○ 20 ptsFind the number of real solutions $x$ of the equation $\sin\left(\frac{\pi\sqrt x}{4}\right)+\cos\left(\frac{\pi\sqrt{2-x}}{4}\right)=\sqrt 2$ .
Solutions of a trigonometric equation ●●○○○ 20 ptsFind the number of solutions to $\frac{1}{\sin^2 \theta} - \frac{1}{\cos^2 \theta} - \frac{1}{\tan^2 \theta} - \frac{1}{\cot^2 \theta} - \frac{1}{\sec^2 \theta} - \frac{1}{\csc^2 \theta} = -3$ in the interval $0 \le \theta \le 2\pi$ .
Real roots of an exponential-cosine equation ●●○○○ 20 ptsFind the number of real roots of the equation $e^{\cos x} - e^{-\cos x} - 4 = 0$ .
Real solutions of x/100 = sin x ●●●○○ 50 ptsFind the number of real solutions of the equation $\frac{x}{100} = \sin x$ .
Removable sine factors keeping natural roots ●●●○○ 50 ptsThe left-hand side of the equation $\sin \frac{\pi}{x} \sin \frac{2\pi}{x} \sin \frac{3\pi}{x} \cdots \sin \frac{2015\pi}{x}=0$ is a product of factors $\sin \frac{n\pi}{x}$ for $n=1,2,\ldots,2015$ . What is the maximum number of these factors that can be crossed out so that the number of natural-number roots of the equation does not change?
Unique solution with a parameter ●●●○○ 50 ptsFind all values of the parameter $a$ for which the equation $a x^{2}+\sin ^{2} x=a^{2}-a$ has a unique solution. Give the value of $a$ .
Intersections of a log and a sine graph ●●●●○ 100 ptsHow many real numbers $x$ satisfy the equation $\frac{1}{5}\log_2 x = \sin (5\pi x)$ ?
Roots in [0,1] after sine substitution ●●●●○ 100 ptsUsing a suitable substitution, determine the number of roots of the equation $8x(1-2x^2)(8x^4-8x^2+1)=1$ that lie in the interval $[0,1]$ .
Real roots of a sine-power equation ●●●●○ 100 ptsFind the number of real roots of the equation $\sin(2013\pi x)=x^{2013}$ .

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