Prove that the trigonometric equation  sin(cosx) = cos(sinx)   has no solutions.  I graphed the functions and I can see they are never equal to eachother, but how do I do a formal proof for this?
26665
The following problem was on a math competition that I participated in at my school about a month ago:      Prove that the equation $\cos(\sin x)=\sin(\cos x)$ has no real solutions.   I will outline my proof below.  I think it has some holes.  My approach to the problem was to say that the following equations must have real solution(s) if the above equation has solution(s):  $$ \cos^2(\sin x)=\sin^2(\cos x)\\ 1-\cos^2(\sin x)=1-\sin^2(\cos x)\\ \sin^2(\sin x)=\cos^2(\cos x)\\ \sin(\sin x)=\pm\cos(\cos x)\\ $$  I then proceeded to split into cases and use the identity $\cos t = \sin(\frac{\pi}{2} \pm t\pm y2\pi)$ to get   $$ \sin x=\frac{\pi}{2} \pm \cos x\pm y2\pi\\ $$  and the identity $-\cos t = \sin(-\frac{\pi}{2}\pm t\pm y2\pi) $ to get   $$ \sin x=- \frac{\pi}{2}\pm \cos x \pm y2\pi.\\ $$  where $y$ is any integer.  I argued that $y=0$ was the only value of $y$ that made any sense (since the values of sine and cosine remain between $-1$ and $1$).  Therefore, the above equations become   $$ \sin x=\frac{\pi}{2} \pm \cos x\implies \sin x \pm \cos x=\frac{\pi}{2}\\ $$  and   $$ \sin x=- \frac{\pi}{2}\pm \cos x\implies \cos x\pm\sin x= \frac{\pi}{2}.\\ $$  Then, by a short optimization argument, I showed that these last two equations have no real solutions.  First, does this proof make sense?  Second, if my proof makes sense, then I feel that it was not very elegant nor simple.  Is my approach the best, or is there a better (i.e., more elegant, shorter, simpler) proof?
117251
I came across this problem when I was at a book store inside of a book made to prepare Berkeley graduates to pass a mandatory exam. I wanted to buy the book, but, alas, I didn't have the money (forty bucks is a lot of money when you don't have a job). So I took my phone out and started taking as many pictures as I could. Unfortunately, I didn't take any pictures of the solutions!  Enough backstory. Time for math. The problem is as follows:  Prove that $\cos^p( \theta) \le \cos(p \theta)$ if $0\le\theta\le\frac\pi2$ and $0\le p\le 1$.  I tried using the series expansion for cosine, but that seemed to be a dead end. Then I tried using Euler's theorem, but I got stuck. Then I got distracted and started to think about other cosine identities. For example, $$\cos\left(\frac\theta2\right)=\pm\sqrt{\frac{1+\cos(\theta)}{2}}.$$ Then what is $\cos(\frac\theta3)$ equal to? I tried to figure it out then I realized that I needed to find the root of a cubic polynomial. Then I realized that I'm better at staying focused than finding the roots of cubic polynomials.  Anyways, a proof would be nice here. I really appreciate any hints or answers. I apologize for my digressions!
92868
Using double angle and compound angles formulae prove,  $$ \frac{1-\cos x}{\sin x}     = \tan\frac{x}{2} $$  Can someone please help me figure this question, I have no idea how to approach it?
80229
How many solutions has the equation $\sin x= \frac{x}{100}$ ?   Usually when I was asked to solve this type of problem, I would solve it graphically but this one seems to be trickier. It doesn't seem wise to put $f(x)=\sin x$ and $g(x)=\frac{x}{100}$ in the same graph and then counting all the intersection points.  What would be some algebraic methods to solve this?
44147
We need to prove that $$\dfrac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac 1{\sec\theta - \tan\theta}$$  I have tried and it gets confusing.
11533
I'm tutoring for a college math class and we're doing putnam problems next week and this one stumped me:      Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers $x$.
59265
I've been wondering, what is the general form for the integral of $\sin(\cos x)$? I tried integrating it myself and couldn't, tried with Wolfram Alpha and it couldn't either. If you think about it conceptually, it should have a general form as it's graph if just like sin and cos, which both have general forms. If it doesn't have a general form, I'd like to know why. Thanks!
3610
Solve $\cos^{n}x-\sin^{n}x=1$ with $n\in \mathbb{N}$  I have no idea how to deal with this crazy question. One idea came into my mine is factorization, but I can't go on... Can anyone help me please? Thank you.
38104
Question: If $\alpha$ and $\beta$ are the solutions of $a\cos \theta + b\sin \theta = c$, then show that: $$\cos (\alpha + \beta) = \frac{a^2 - b^2}{a^2 + b^2}$$  No idea how to even approach the problem. I tried taking two equations, by substituting $\alpha$ and $\beta$ in place of $\theta$ in the equation and manipulating them, but that didn't get me anywhere. Please help!
114279
Prove that for $ 0&lt;x&lt; \pi $, $$ \quad S_n(x) = \sin x + \frac{ \sin3x }{3} + ... + \frac{ \sin((2n-1)x) }{2n-1} &gt;0 \quad \forall n = 1,2,... $$  Having trouble with this problem. This is an olympiad-style question, so an answer that doesn't use calculus or analysis would be preferred. A possible approach is induction, but for this we need to find a function in terms of $n$ and $x$ so that we can actually use the inductive step. If anyone has any ideas they would be appreciated.  If you really want to go down the calculus route (at this point I don't mind), then $ S_n' (x) = \cos x + \cos 3x + ... +\cos((2n-1)x) $ , which you can find a closed form for, but I don't know how useful that is.
113327
So, how do I prove that this trigonometric sum equals to zero?   $$\sum_{k = 0}^{n - 1} \cos \left(\frac{2\pi k}{n}\right) = 0$$
104619
Find all the solutions of $$\sin^5x+\cos^3x=1$$   Trial:$x=0$ is a solution of this equation. How can I find other solutions (if any). Please help.
3785
I should prove this trigonometric identity. I think I should get to this point : $\cos(3x) = 4\cos^3 x - 3\cos x $ But I don't have any idea how to do it (I tried solving $\cos(x+60^{\circ})\cos(x-60^{\circ})$ but I got nothing)
82453
Prove that $\cos\theta$ is root of equation $8x^3-4x^2-4x+1=0$, given $\theta=\frac{\pi}{7}$.   I put $\cos\theta$ in equation, but couldn't show the left-hand side to be zero.
20656
I have a trigonometric function and I'm interested to know whether or not it has a period. At this stage I'm fairly certain that it is not periodic. However, I don't know how to prove it. Can anyone help?  This is the function:  $$g(x) = \sin(2 \pi (x-a) \times b \times \cos(2 \pi (x - a) \times c)) + d$$  Thanks
86531
The question is  Find the solutions to the equation $$2\tan(2x)=3\cot(x) , \space 0&lt;x&lt;180$$  I started by applying the tan double angle formula and recipricoal identity for cot $$2* \frac{2\tan(x)}{1-\tan^2(x)}=\frac{3}{\tan(x)}$$ $$\implies 7\tan^2(x)=3 \therefore x=\tan^{-1}\left(-\sqrt\frac{3}{7} \right)$$ $$x=-33.2,33.2$$ Then by using the quadrants  I was lead to the final solution that $x=33.2,146.8$ however the answer in the book has an additional solution of $x=90$, I understand the reasoning that $\tan(180)=0$ and $\cot(x)$ tends to zero as x tends to 90 however how was this solution found? Is there a process for consistently finding these &quot;hidden answers&quot;?
41148
I'm doing some math exercises but I got stuck on this problem.  In the book of Bogoslavov Vene it says to prove that:  $$\frac{\sin x}{1-\cos x} = \frac{1+\cos x}{\sin x}.$$  It is easy if we do it like this: $\sin^2 x = (1-\cos x)(1+\cos x)=1-\cos^2 x$. But how to prove it just by going from the left part or from the right  Can anybody help me?  Thank you!