cos(x+y) cos(x-y) = cos^2(x) - sin^2(y)?

Question

cos(x+y) cos(x-y) = cos^2(x) - sin^2(y)?

Answer 1

Remember the cosine of a sum formula



cos(x + y) = cosx * cosy - sinx * siny

cos(x - y) = cosx * cosy + sinx * siny



cos(x+y) cos(x-y) = [cosx * cosy - sinx * siny] [cosx * cosy + sinx * siny]



= cos^2x cos^2 y - sinx siny cosx cosy + sinx siny cosx cosy - sin^2x sin^2y

= cos^2 x cos^2 y - sin^2 x sin^2 y



Note that since sin^2 x + cos^2 y = 1 and sin^2 y + cos^2 y = 1 we have cos^2 y = 1 - sin^2 y and sin^2 x = 1 - cos^2 x



cos^2 x (1 - sin^2 y) - (1 - cos^2 x) sin^2 y

cos^2 x - sin^2 y cos^2 x - sin^2 y + sin^2 y cos^2 x

cos^2 x - sin^2 y



Thus we are done!

Answer 2

the left is [cx cy - sx sy][cxcy+sxsy] = cx²cy² - sx² sy² = cx²[1-sy²] - sx²sy² = cx² - cx²sy² - sx² su²



= cx² - sy² [ cx² + sx²] = cx² - sy²

Answer 3

Y Cos X

Answer 4

LHS

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

= [ cos x cos y - sin x sin y] [ cos x cos y + sin x sin y]

= 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

= RHS

Answer 5

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

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

so

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

Now you are done... you have proven

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