If $x+y+z=\pi$ how to prove that $\sin x+\sin y+\sin z=4\cos\frac{x}{2}\cos\frac{y}{2}\cos\frac{z}{2}$?  I got that $\sin x+\sin y+\sin z=2\sin\frac{x+y}{2}\cos\frac{x-y}{2}+\sin x\cos y+\sin y\cos x$ and I don't know what to do next.   Can somebody help me, please?
120484
Assume: $\alpha + \beta + \gamma = \pi$ (Say, angles of a triangle)  Prove: $\sin\alpha + \sin\beta + \sin\gamma = 4\cos{\frac{\alpha}{2}}\cos{\frac{\beta}{2}}\cos{\frac{\gamma}{2}}$    , however I want to avoid using the sum-to-product identity because technically the book I go by hasn't covered it yet.  So, is there a way to prove it with identities only as advanced as $\sin\frac{\alpha}{2}$?    Edit: Just giving a hint will probably be adequate (i.e. what identity I should manipulate).
72973
I have to prove the following statement.     $$1+\cos{2\pi\over5}+\cos{4\pi\over5}+\cos{6\pi\over5}+\cos{8\pi\over5}=0$$   I have tried to use the sum of angles formula for cosine, but didn't get to a point where I'd be able to show that it is equal to $0$.
10368
Prove by induction:   $$\sin{x}+\sin{3x}+\dots+\sin{(2n-1)x}=\frac{\sin^2{nx}}{\sin{x}}$$   I tried the problem using the normal rule of induction(the first principle), but I failed.I failed to make the form $\sin^2{(m+1)x}$. Somebody help me.
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.
59687
Prove $\sin(\pi/2)=1$ using the Taylor series definition of $\sin x$,   $$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$$   It seems rather messy to substitute in $\pi/2$ for $x$. So we have  $$\sin(\pi/2)=\sum_{n=0}^{\infty} \frac{(-1)^n(\pi/2)^{2n+1}}{(2n+1)!}.$$  I'm not too sure where to go from here.  Any help would be appreciated! Thanks!
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!
29025
Show that if $x, y,$ and $z$ are consecutive terms of an arithmetic sequence, and $\tan y$ is defined, then  $$\frac{\sin x + \sin y + \sin z }{\cos x + \cos y + \cos z} = \tan y. $$  I'm not sure what trig identities I would use and how to use them.  Could I get some help?  Thanks.
77505
Could anyone offer a proof that  $$ \cos^2(x)\cosh^2(y) + \sin^2(x)\sinh^2(y) = -1 + \sin^2(x) - \sinh^2(y)? $$
79834
if $x+2y+z=4$ and $x,y,z$ are real number. then find maximum value of $xy+yz+zx$  putting $x+z=4-2z$ in $y(x+z)+zx = y(4-2z)+zx = 4y-2yz+zx$  i wan,t be able to  go further,could some help me with this
57873
Is there an easy way to prove the identity?     $$\cos \left ( \frac{\pi}{7} \right ) - \cos \left ( \frac{2\pi}{7} \right ) + \cos \left ( \frac{3\pi}{7} \right ) = \cos \left (\frac{\pi}{3} \right )$$   While solving one , I am stuck, which looks obvious but without any feasible way to approach.   Few observations, not sure if it would help $$ \begin{align} \dfrac{\dfrac{\pi}{7}+\dfrac{3\pi}{7}}{2} &amp;= \dfrac{2\pi}{7}\\\\ \dfrac{\pi}{7} + \dfrac{3\pi}{7} + \dfrac{2\pi}{7} &amp;= \pi - \dfrac{\pi}{7} \end{align} $$
75606
If z and w are complex numbers such that $|z+w|$ = $|z-w|$, prove that $\arg(z)-\arg(w)= \pm\pi/2$.   Can this be solved algebraically or would a graphic interpretation be better. Both methods would be grateful.
45926
Find the value of $\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}$  My approach :   I used $\sin A +\sin B = 2\sin(A+B)/2\times\cos(A-B)/2 $  $\Rightarrow \sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7} = 2\sin(3\pi)/7\times\cos(\pi/7)+\sin(8\pi/7) $  $= 2\sin(3\pi)/7\times\cos(\pi/7)+\sin(\pi + \pi/7) $  $= 2\sin(3\pi)/7\times\cos(\pi/7)-\sin(\pi/7) $  I am not getting any clue how to proceed further or whether it is correct or not. Please help thanks..
28096
How does one solve this equation?     $$\cos {x}+\sin {x}-1=0$$   I have no idea how to start it.  Can anyone give me some hints? Is there an identity for $\cos{x}+\sin{x}$?  Thanks in advance!
74361
Ok, so I have the following proof. Let $z$ and $w$ be complex numbers. Prove $\lvert z+w \rvert ^2 + \lvert z-w \rvert^2 = 2[\lvert z \rvert^2 + \vert w \rvert^2]$. Using $\vert z \rvert^2=z\bar{z}$, I have said   $\lvert z+w \rvert ^2 + \lvert z-w \rvert^2 = 2(\lvert z \rvert^2) + \vert w \rvert^2)$  $(z+w)\bar{(z+w)}+(z-w)(\bar{(z-w)}$  $(z+w)(z-w)+(z-w)(z+w)$  $z^2 - w^2 + z^2 - w^2$  $z^2 + z^2 + -w^2 + -w^2$  $2(z^2) + 2(-w^2)$  $2[z^2 + (-w)^2]$  $2[\lvert z \rvert^2 + \vert w \rvert^2]$  I guess my question is, is my logic correct? I am not so skilled with proof writing and though I feel that this is correct, I have been known to be wrong in the past. Just need some insight. Thank you.
12287
$$\int_0^{2\pi} \frac{1}{2+\sin(x)} \ dx$$  $$\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$$  $$z=\alpha(x)=e^{ix}$$   $$\sin(x)=\frac{z^2-1}{2zi}$$ $$2+\sin(x)=\frac{4zi+z^2-1}{2zi}$$ $$\frac{1}{2+\sin(x)}=\frac{2zi}{4zi+z^2-1}$$    $$\int_\alpha \frac{2zi}{4zi+z^2-1} \ \ \frac{1}{zi}$$    $$4zi+z^2-1=0$$ $$z_1=\frac{-4i+2\sqrt{3}i}{2}$$ $$z_2=\frac{-4i-2\sqrt{3}i}{2}$$    $$\rvert z_1 \rvert &lt;1$$ $$\rvert z_2 \rvert &gt;1$$   Using Residue theorem:   $$\int_\alpha \frac{2}{4zi+z^2-1}=2 \pi i \ \lim_{z\rightarrow-2i+\sqrt{3}i} \ \frac{2}{z+2i+\sqrt{3}i}=\frac{2 \pi}{\sqrt{3}}$$    Is it correct?  Thanks!
38403
How would I verify the following double angle identity. $$ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $$ So far I have done this. $$ (\sin A\cos B+\cos A\sin B)(\sin A\cos B-\cos A\sin B)  $$But I am not sure how to proceed.
92295
If $$(1+ \cos A) (1+ \cos B) (1+\cos C)= y = (1- \cos A) (1-\cos B) (1-\cos C)$$ then prove that $$y = \pm \sin A \sin B \sin C$$