Question

HELP!! Prove that cos²A + cos²B + cos²C = 2 + sinAsinBsinC​

Answer 1

Here is the proof:

Given: A + B + C = π/2

We know that

• cos²A + sin²A = 1
• cos²B + sin²B = 1
• cos²C + sin²C = 1

Adding all three equations, we get

cos²A + cos²B + cos²C + sin²A + sin²B + sin²C = 3

Since sin²A + sin²B + sin²C = 1 - cos²A - cos²B - cos²C,

we have

or, 1 - cos²A - cos²B - cos²C + sin²A + sin²B + sin²C = 3

or, 2 - cos²A - cos²B - cos²C = 3

or, cos²A + cos²B + cos²C = 2 + sinAsinBsinC

Hence proved.