Answer:
√2/2
Step-by-step explanation:
cos(45 - x)cos(x) - sin(45 -x)sin(x)
Using trigonometric identities,
cos(A - B) = cosAcosB + sinAsinB where A = 45 and B = x. So,
cos(45 - x) = cos45cosx + sin45sinx
= (1/√2)cosx + (1/√2)sinx (since sin45 = cos45 = 1/√2)
Also,
sin(A - B) = sinAcosB - cosAsinB where A = 45 and B = x. So,
sin(45 - x) = sin45cosx - cos45sinx
= (1/√2)cosx - (1/√2)sinx (since sin45 = cos45 = 1/√2)
So.
cos(45 - x)cos(x) - sin(45 -x)sin(x) = [(1/√2)cosx + (1/√2)sinx]cos(x) - [(1/√2)cosx - (1/√2)sinx]sin(x)
= (1/√2)cos²x + (1/√2)sinxcos(x) - [(1/√2)sinxcosx - (1/√2)sin²x]
= (1/√2)cos²x + (1/√2)sinxcos(x) - (1/√2)sinxcosx + (1/√2)sin²x
= (1/√2)cos²x + (1/√2)sin²x
= (1/√2)[cos²x + sin²x]
= (1/√2) (since cos²x + sin²x = 1)
= 1/√2 × √2/√2
= √2/2