query: How to prove $\big | e^{it}-1 \big |=2 \bigg |\sin \left(\frac{t}{2}\right) \bigg |$
positive_passages.docid: 83359
positive_passages.text: Complex number identity by trigonometry
negative_passages.docid: 121548
negative_passages.text: proof of an inequality $ \left( 1+ \frac{x}{p} \right) ^{p}$
negative_passages.docid: 89429
negative_passages.text: Prove $\sup \left| f'\left( x\right) \right| ^{2}\leqslant 4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) \right| $
negative_passages.docid: 57203
negative_passages.text: Prove that $\lim_{x \to \infty}\big(\frac{x}{x-1}\big)^x$ is also $e$.
negative_passages.docid: 26211
negative_passages.text: Computing $\lim_{x \to \infty} x \biggl[ \frac{1}{e} - \left( \frac{x}{x+1} \right)^x \biggr]$
negative_passages.docid: 34398
negative_passages.text: Prove $\left| \int_a^b f(t) dt \right| \leq \int_a^b \left| f(t) \right| dt$
negative_passages.docid: 86348
negative_passages.text: Proving $\sum\limits_{k=0}^{n-1} \Bigl[x + \frac{k}{n}\Bigr] = [nx]$
negative_passages.docid: 66082
negative_passages.text: Using $\sqrt{1-t}\leq 1-\frac t2$ to show that $\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots2n}\geq\frac1{2\sqrt n}$
negative_passages.docid: 124208
negative_passages.text: Proof $|\sin(x) - x| \le \frac{1}{3.2}|x|^3$
negative_passages.docid: 78174
negative_passages.text: How prove this inequality $\left(\int_{0}^{1}f(x)dx\right)^2\le\frac{1}{12}\int_{0}^{1}|f'(x)|^2dx$
negative_passages.docid: 64277
negative_passages.text: How to prove ${{a}^{a}}{{b}^{b}}\ge {{\left(\frac{a+b}{2}\right)}^{a+b}}$ ?thanks.
negative_passages.docid: 71708
negative_passages.text: Proving that $\left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) \lvert f(z) \rvert^2 = 4 \lvert f'(z) \rvert^2$
negative_passages.docid: 105641
negative_passages.text: Prove that $ab\leq\frac{1}{2\varepsilon}a^2+\frac{\varepsilon}{2}b^2$
negative_passages.docid: 21380
negative_passages.text: Proof for $e^z = \lim \limits_{x \rightarrow \infty} \left( 1 + \frac{z}{x} \right)^x$
negative_passages.docid: 48503
negative_passages.text: Proving : $ \bigl(1+\frac{1}{n+1}\bigr)^{n+1} \gt (1+\frac{1}{n})^{n} $
negative_passages.docid: 127759
negative_passages.text: Prove that $\frac{\sin(a)}{\sin(b)} < \frac{a}{b} < \frac{\tan(a)}{\tan(b)}$ where $0 < b < a < \frac{\pi}{2}$
negative_passages.docid: 57670
negative_passages.text: Show that $|z_1 + z_2|^2 < (1+C)|z_1|^2 + \left(1 + \frac{1}{C}\right) |z_2|^2$