Prove the identity

Question

sin(x+y+z)+sin(x+y-z)+sin(x-y+z)+ sin(x-y-z) = 4 sin(x)cos(y)cos(z)

This identity is so long and after i tried to expand the left side and it just looked something crap

Thanks for you help :)

Answer 1

LS
= sin((x+y)+z)+sin((x+y)-z)+sin((x-y)+z)+ sin((x-y)-z)
= sin(x+y)cosz + cos(x+y)sinz
+ sin(x+y)cosz - cos(x+y)sinz
+ sin(x-y)cosz + cos(x-y)sinz
+ sin(x-y)cosz - cos(x-y)sinz

= 2sin(x+y)cosz + 2sin(x-y)cosz
= 2[cosz(sinxcosy + cosxsiny)] + 2[cosz(sinxcosy - cosxsiny)]

= 2sinxcosycosz + 2sinycosxcosz + 2sinxcosycosz - 2 sinycosxcosz

= 4 sinxcosycosz

= RS

Answer 2

I once had a great geometry teacher back in high school in Seattle. He was old and close to retirement. He used to say, to a mostly bored and unappreciative class: "Try to see the beauty in it." A few of us did. It was my favorite subject. Trig was next.

Reiny's proof reminds me of that beauty.

Answer 3

Well, i tried to see the beauty in trigonometry but to me, it is just too HARD!!!!!!!!

There r so many formulae in trigonometry and how to i know which one to use.

After i stay in the desk for 15 mins, i just wanna throw this stupid book away. I cant gain anything even i try so hard.

Answer 4

Thanks for the comment.

Back in the days when we still wrote on a blackboards with chalk and we still had care-takers that would clean those boards at the end of the day....
We once did a proof of "in any quadrilateral the largest area is obtained when opposite angles are supplementary" and it filled about 3 sections of blackboard.
After it was done, the students drew a large picture frame around it.
Next day it was still there with a note from the care-taker.
"I looked at this, did not understand anything about it, but it sure looks like a piece of art"

Your comment reminded me of that day, thanks.

Answer 5

A ladder that is 6 meters long is placed against a wall. It makes an angle 0f 34 degrees with the wall. Find how high up the wall it reaches and the distnce the base of the ladder is away from the wall