Proving Triangle Angles Equation: cos^2 A + cos^2 B + cos^2 C + 2*cosA*cosB*cosC

[ER901]

Homework Statement

I need to prove the following equation:
cos^2 A + cos^2 B + cos^2 C + 2*cosA*cosB*cosC = 1
where A,B,C are the angles of a triangle

Homework Equations

A+B+C = 180

The Attempt at a Solution

I substituted the sum of A,B,C into the equation, and now I have something like
... + cos^2(180-A-C)+... so on, but I don't know what to do next.

Thanks

 

[Gib Z]

Welcome to Physicsforums ER901!

How about using the Cosine rule to express the Cosines as functions of the sides? Theres some messy algebra but it should work out fine.

 

[Architeuthis Dux]

for your question! Let's start by breaking down the equation and understanding what each term represents. The left side of the equation is a sum of three cosine squared terms and a product of three cosine terms. The right side is simply the number 1.

To prove this equation, we will use the fact that the sum of the angles in a triangle is always 180 degrees. This is represented by the equation A+B+C=180.

First, let's rewrite the equation in terms of A, B, and C using the substitution A=180-B-C. This gives us:

cos^2(180-B-C) + cos^2(B) + cos^2(C) + 2*cos(180-B-C)*cos(B)*cos(C) = 1

Next, we can use the identity cos(180-x)=-cos(x) to rewrite the first term as:

cos^2(B+C) + cos^2(B) + cos^2(C) - 2*cos(B+C)*cos(B)*cos(C) = 1

Now, let's use the double angle identity cos(2x)=2cos^2(x)-1 to rewrite the first and third terms as:

(2*cos^2(B)*cos^2(C) - 1) + cos^2(B) + cos^2(C) - (2*cos(B)*cos(C)*cos(B+C)) = 1

Simplifying this, we get:

2*cos^2(B)*cos^2(C) + cos^2(B) + cos^2(C) - 2*cos(B)*cos(C)*cos(B+C) = 2

Using the identity cos(x+y)=cos(x)*cos(y)-sin(x)*sin(y), we can rewrite the last term as:

2*cos^2(B)*cos^2(C) + cos^2(B) + cos^2(C) - 2*(cos(B)*cos(B)*cos(C) - sin(B)*sin(C)*cos(B)*cos(C)) = 2

Simplifying this, we get:

2*cos^2(B)*cos^2(C) + cos^2(B) + cos^2(C) - 2*cos^2(B)*cos^2(C) + 2*sin(B)*sin(C)*cos(B)*cos(C) = 2

Now, we can use the Pythagorean identity sin^2(x)+cos^2(x)=