cos(x+y) cos(x-y)=. . . .

cos^2(x) - sin^2(y)

Pembahasan

Untuk menjawab soal ini, kita dapat menggunakan identitas trigonometri:
cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
cos(x-y) = cos(x)cos(y) + sin(x)sin(y)

Maka, cos(x+y) cos(x-y) = (cos(x)cos(y) - sin(x)sin(y))(cos(x)cos(y) + sin(x)sin(y))
Ini adalah bentuk (a-b)(a+b) = a^2 - b^2, dengan a = cos(x)cos(y) dan b = sin(x)sin(y).
Jadi, cos(x+y) cos(x-y) = (cos(x)cos(y))^2 - (sin(x)sin(y))^2
= cos^2(x)cos^2(y) - sin^2(x)sin^2(y)

Kita juga bisa menyederhanakannya lebih lanjut menggunakan identitas sin^2(x) + cos^2(x) = 1.
cos^2(x)cos^2(y) - sin^2(x)sin^2(y)
= cos^2(x)(1-sin^2(y)) - (1-cos^2(x))sin^2(y)
= cos^2(x) - cos^2(x)sin^2(y) - sin^2(y) + cos^2(x)sin^2(y)
= cos^2(x) - sin^2(y)

Atau, bisa juga diubah menjadi:
cos^2(x)cos^2(y) - sin^2(x)sin^2(y)
= (1-sin^2(x))cos^2(y) - sin^2(x)(1-cos^2(y))
= cos^2(y) - sin^2(x)cos^2(y) - sin^2(x) + sin^2(x)cos^2(y)
= cos^2(y) - sin^2(x)

Jadi, cos(x+y) cos(x-y) = cos^2(x) - sin^2(y) = cos^2(y) - sin^2(x).