Could somebody please help me prove this identity? thanks!!!?

Question

cos(45degrees+x)+cos(x+315degrees)=square root of 2 cos x

Answer 1

cos (45 + x) + cos (x + 315) = sqrt(2)cos(x)

(Side note: 45 degrees = pi/4)
(315 degrees = 7pi/4)

Whenever solving identities, you choose either the left hand side (LHS) or the right hand side (RHS). The best one to choose is the more complex side, and in this case it's the LHS.

LHS = cos (45 + x) + cos (x + 315)

Remember the cosine addition identity:
cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
This is what we apply.

LHS = [cos(45)cos(x) - sin(45)sin(x)] + [cos(x)cos(315) - sin(x)sin(315)]

Since those are familiar values on the unit circle, we can evaluate them directly.

LHS = [sqrt(2)/2]cos(x) - [sqrt(2)/2]sin(x) + cos(x)[sqrt(2)/2] -
sin(x)[-sqrt(2)/2]

Simplifying this a bit more and doing a bit of rearranging, we have

LHS = [sqrt(2)/2]cos(x) - [sqrt(2)/2]sin(x) + [sqrt(2)/2]cos(x)
+ [sqrt(2)/2]sin(x)

{Note: the minus sign in front of the last sqrt(2)/2 merged with the minus sign on the outside to form a plus}.

Notice that the second and fourth terms cancel each other out (one being a plus, the other being a minus, and them being exactly the same).

LHS = [sqrt(2)/2]cos(x) + [sqrt(2)/2]cos(x)

Also notice that these two terms are exactly the same, and we have two of them.

LHS = 2 [sqrt(2)/2]cos(x)

Now, notice the 2 cancels out with the 2 on the denominator, and our result is equal to the right hand side.

LHS = [sqrt(2)]cos(x) = RHS

Answer 2

Let's use the addition formula for the cosine:
cos(a + b) = cos a cos b - sin a sin b
which easily implies
cos(a - b) = cos a cos b + sin a sin b.
Now note that
cos(360 - x) = cos 360 cos x + sin 360 sin x = cos x
cos(x + 315)= cos(360-(315 + x)) = cos(45 -x)
To finish the job we have
cos(45 +x) = cos 45 cos x - sin 45 sin x
cos(45 -x) = cos 45 cos x + sin 45 sin x.
Adding these, we get
2cos 45 cos x = sqrt(2)*cos x.
Hope that helps!

Answer 3

dear tink

just make use twice of the formule

cos ( a+b) = cos a cos b - sen a sen b and remembering

2 cos 45 = 2 cos 315 = root of 2, will get what you wish.

Answer 4

cos(45+x)+cos(x+315)=RHS
LHS=
cos45cosx-sin45sinx
+
cosxcos315-sinxsin315
remember cos315=cos(-45)=cos45
sin315=sin(-45)= - sin45

Hence
LHS=cos45cosx-sin45sinx
+
cosx. cos45+sinx. sin45
=2cosxcos45 + 0
=2 root(2)/2 .cosx
=root(2). cosx
=RHS

Answer 5

cos(45+x) + cos(x+315)
= cos(45+x) + cos(x-45)
= 2cos(45)cos x
=√2 cos x

Answer 6

Can you add additional details? The question is truncated so maybe you can write it in pieces so someone can help you.

>> sahsjing's proof works

Answer 7

Just a plumber next have a HAPPY NEW YEAR

Answer 8

no.