trig help please?

Question

please help me solve this identity

cos(α+β)cos(α-β)=cos²β-sin²-α

thankyou

Answer 1

Prove the identity.

cos(α + β)cos(α - β) = cos²β - sin²(-α)
___________

Left Hand Side = cos(α + β)cos(α - β)

= [(cosα)(cosβ) - (sinα)(sinβ)] [(cosα)(cosβ) + (sinα)(sinβ)]

= (cos²α)(cos²β) - (sin²α)(sin²β)

= (1 - sin²α)(cos²β) - (sin²α)(1 - cos²β)

= cos²β - (sin²α)(cos²β) - sin²α + (sin²α)(cos²β)

= cos²β - sin²α = cos²β - sin²(-α) = Right Hand Side