Question

Find the value of the following expression:

cos 60° cos 45° – sin 60° sin 45°

Answer 1

Given: cos 60° cos 45° – sin 60° sin 45° (Use the cosine sum identity: cos (A + B) = cos A cos B – sin A sin B).

Step-wise calculation:

1. Apply the identity with A = 60°, B = 45°:

cos 60° cos 45° – sin 60° sin 45°

= cos (60° + 45°)

= cos 105°

2. Substitute exact values:

`cos 60^circ = 1/2` 

`cos 45^circ = sqrt(2)/2` 

`sin 60^circ = sqrt(3)/2` 

`sin 45^circ = sqrt(2)/2`

3. Compute the products:

`(1/2)(sqrt(2)/2) - (sqrt(3)/2)(sqrt(2)/2)`

= `sqrt(2)/4 - sqrt(6)/4`

= `((sqrt(2) - sqrt(6)))/4`

cos 60° cos 45° – sin 60° sin 45°

= `((sqrt(2) - sqrt(6)))/4`

= `-((sqrt(6) - sqrt(2)))/4` 

= – 0.258819