Given $\int_{0}^{\pi} f(\theta)\cos\theta \,d\theta=\int_{0}^{\pi} f(\theta)\sin\theta \,d\theta ,$ prove at least two zero points.
82567
Prove that $ f(x) $ has at least two real roots in $ (0,\pi) $
103706
If $\{f(0)\}^{2} + \{f'(0)\}^{2} = 4$ then there is a $c$ with $f(c) + f''(c) = 0$
12966
Riemann Sums proving function identically zero
152336
Any hint to solve given integral $\int_0^{2{\pi}}{{d\theta}\over{a^2\cos^2\theta+b^2\sin^2\theta}}$?
71071
Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$
69000
Looking for closed-forms of $\int_0^{\pi/4}\ln^2(\sin x)\,dx$ and $\int_0^{\pi/4}\ln^2(\cos x)\,dx$
134043
Prove that for $ f(x+2\cdot\pi)=f(x), x \in \mathbb{R} $, there exists $ x_0 $ so $ f(x_0+\pi)=f(x_0) $
51837
Show that $\int_{0}^{\pi} xf(\sin(x))\text{d}x = \frac{\pi}{2}\int_{0}^{\pi}f(\sin(x))\text{d}x$
53307
how to strictly prove $\sin x<x$ for $0<x<\frac{\pi}{2}$
7388
Evaluate: $\int_0^{\pi} \ln \left( \sin \theta \right) d\theta$
127284
Closed form of $\int_0^{\pi/2} \frac{\arctan^2 (\sin^2 \theta)}{\sin^2 \theta}\,d\theta$
36701
$\displaystyle\int_{0}^{2\pi}\frac{1}{a+ \cos\theta}\,d\theta$
36586
Evaluating $\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{1-2t\cos\theta +t^2}d\theta$
71742
integrate $\int_0^{2\pi} e^{\cos \theta} \cos( \sin \theta) d\theta$
27910
How to evaluate $\int_0^{2\pi} \frac{d\theta}{A+B\cos\theta}$?
124460
Show that if $f \in \mathbb{C} \left[x_1, \dots, x_n \right]$ vanishes at every point of $\mathbb{Z}^n$, then $f$ is the zero polynomial.
2414
If $f \in \operatorname{Hol}(D)$, $f(\frac{1}{2}) + f(-\frac{1}{2}) = 0$, prove that $|f(0)| \leq \frac{1}{4}$