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.