Question

Prove that: cos^2 (45+A)+cos^2 (45-A)=1​

Answer 1

Explanation:

cos 2x=cos²x-sin²x=cos²x-(1-cos²x)=cos²x-1+cos²x=2cos²x-1

2cos²x=1+cos2x

`cos^2x=(1)/(2)(1+cos2x)`

cos²(45+A)+cos²(45-A)

`=(1)/(2)(1+cos(90+2A))+(1)/(2)(1+cos(90-2A))\\=(1)/(2) (1-sin2A)+(1)/(2) (1+sin 2A)\\=(1)/(2) (1-sin2A+1+sin 2A)\\=(1)/(2) *2\\=1`

cos (90-x)=sin x

cos (90+x)=-sin x

Answer 2

see explanation

Explanation:

Using the cosine addition formula

cos(A ± B ) = cosAcosB ∓ sinAsinB

Then considering the left side

cos²(45 + A) + cos²(45 - A)

= [ cos45cosA - sin45sinA ]² + [cos45cosA + sin45sinA]]²

= [
`(1)/(√(2) )` cosA -
`(1)/(√(2) )` sinA ]² + [
`(1)/(√(2) )` cosA +
`(1)/(√(2) )` sinA ]²

=
`(1)/(2)`cos²A - sinAcosA +
`(1)/(2)` sin²A +
`(1)/(2)` cos²A + sinAcosA +
`(1)/(2)` sin²A

= cos²A + sin²A

= 1

= right side , then proven