cos^2xcos^2y+sin^2xsin^2y+sin^2xcos^2y+sin^2ycos^2x = 1

Where are you getting stuck?

I got up to 1-sin^2x - sin^2y + sin^2xsin^y +sin^2x-sin^2xsin^2y + sin^2y-sin^2ysin^x. I don't know if that's right or wrong.

To prove that cos^2(x)cos^2(y) + sin^2(x)sin^2(y) + sin^2(x)cos^2(y) + sin^2(y)cos^2(x) equals 1, we can use the concept of the trigonometric identity: sin^2(x) + cos^2(x) = 1.

Let's start by expanding the given expression:

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

Now, let's rearrange the terms:

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

We can write the first part of the expression as follows:

cos^2(y) * (cos^2(x) + sin^2(x))

Using the trigonometric identity mentioned earlier, cos^2(x) + sin^2(x) equals 1. So, we can replace it with 1:

cos^2(y) * 1

Since any number multiplied by 1 is the number itself, we have:

cos^2(y)

Next, for the second part of the expression, we can rearrange it as:

sin^2(x) * (sin^2(y) + cos^2(y))

Again, using the trigonometric identity sin^2(y) + cos^2(y) = 1, we can simplify further:

sin^2(x) * 1

Again, any number multiplied by 1 is the number itself:

sin^2(x)

Now, we have cos^2(y) + sin^2(x), which is equivalent to 1, according to the trigonometric identity.

Therefore, the entire expression simplifies to:

cos^2(y) + sin^2(x) = 1

Hence, we have proved that cos^2(x)cos^2(y) + sin^2(x)sin^2(y) + sin^2(x)cos^2(y) + sin^2(y)cos^2(x) equals 1.