query: Prove $(\cos \alpha + i \sin \alpha)^n = \cos ( n \alpha ) + i \sin (n \alpha)$ by induction
positive_passages.docid: 10424
positive_passages.text: Review of a proof for De Moivre's theorem using mathematical induction
negative_passages.docid: 54595
negative_passages.text: Prove $\sum\limits_{i=2}^{n}\frac{1}{(i-1)i}$ = $\frac{(n-1)}{n}$ using induction.
negative_passages.docid: 11711
negative_passages.text: Prove trigonometry identity for $\cos A+\cos B+\cos C$
negative_passages.docid: 149447
negative_passages.text: Proving $\sin (x)=\cos (90^\circ-x)$
negative_passages.docid: 54175
negative_passages.text: Prove the trigonometric identity $\sin^4{x} = \frac{3-4\cos{2x}+\cos{4x}}{8}$
negative_passages.docid: 60230
negative_passages.text: Prove that $\frac{1+\sin\theta+i\cos\theta}{1+\sin\theta-i\cos\theta}=\sin\theta+i\cos\theta$
negative_passages.docid: 73538
negative_passages.text: solving $\cos z + \sin z = i$
negative_passages.docid: 48824
negative_passages.text: Prove: $ \frac{1}{\sin 2x} + \frac{1}{\sin 4x } + \cdots + \frac{1 }{\sin 2^n x} = \cot x - \cot 2^n x $
negative_passages.docid: 25054
negative_passages.text: Proving $\frac{1}{n+1} + \frac{1}{n+2}+\cdots+\frac{1}{2n} > \frac{13}{24}$ for $n>1,n\in\Bbb N$ by Induction
negative_passages.docid: 14435
negative_passages.text: Show by induction: $(1+\frac{1}{n})^{n}<n$
negative_passages.docid: 67527
negative_passages.text: Proof by Induction that $3^n ≥ 1+2^n$
negative_passages.docid: 81775
negative_passages.text: Prove $a^\alpha b^{1-\alpha} \le \alpha a + (1 - \alpha)b, \; a,b > 0,\; 0 < \alpha < 1$
negative_passages.docid: 42285
negative_passages.text: Prove by mathematical induction.
negative_passages.docid: 13291
negative_passages.text: Proving $\sum_{k=1}^n\frac1{\sqrt k}<2\sqrt n$ by induction
negative_passages.docid: 108698
negative_passages.text: proof using the mathematical induction
negative_passages.docid: 14386
negative_passages.text: Proof of an equality involving cosine $\sqrt{2 + \sqrt{2 + \cdots + \sqrt{2 + \sqrt{2}}}}\ =\ 2\cos (\pi/2^{n+1})$
negative_passages.docid: 141371
negative_passages.text: Prove that $\alpha^n+\beta^n+\gamma^n \equiv 4^n+5^n+(-6)^n \pmod{17}$