cos(a+b)cos(a-b)-sin(a+b)sin(a-b)
Using the formula for the cosine of the difference of two angles and the formula for the sine of the sum of two angles, we can simplify the expression:
cos(a+b)cos(a-b)-sin(a+b)sin(a-b= (cos(a)cos(b) - sin(a)sin(b))(cos(a)cos(b) + sin(a)sin(b)) - (sin(a)cos(b) + cos(a)sin(b))(cos(a)sin(b) - sin(a)cos(b)= cos^2(a)cos^2(b) - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos(a)^2cos(b)^2 + sin(a)^2sin(b)^2 - sin(a)^2cos(b)^2 - cos(a)^2sin(b)^= cos(a)^2cos(b)^2 + sin(a)^2sin(b)^2 - sin(a)^2cos(b)^2 - cos(a)^2sin(b)^= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos(a)^2cos(b)^2 + sin(a)^2sin(b)^2 - sin(a)^2cos(b)^2 - cos(a)^2sin(b)^= cos(a)^2cos(b)^2 + sin(a)^2sin(b)^2 - sin(a)^2cos(b)^2 - cos(a)^2sin(b)^= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b= 1
Therefore, the simplified expression is 1.
cos(a+b)cos(a-b)-sin(a+b)sin(a-b)
Using the formula for the cosine of the difference of two angles and the formula for the sine of the sum of two angles, we can simplify the expression:
cos(a+b)cos(a-b)-sin(a+b)sin(a-b
= (cos(a)cos(b) - sin(a)sin(b))(cos(a)cos(b) + sin(a)sin(b)) - (sin(a)cos(b) + cos(a)sin(b))(cos(a)sin(b) - sin(a)cos(b)
= cos^2(a)cos^2(b) - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos(a)^2cos(b)^2 + sin(a)^2sin(b)^2 - sin(a)^2cos(b)^2 - cos(a)^2sin(b)^
= cos(a)^2cos(b)^2 + sin(a)^2sin(b)^2 - sin(a)^2cos(b)^2 - cos(a)^2sin(b)^
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos(a)^2cos(b)^2 + sin(a)^2sin(b)^2 - sin(a)^2cos(b)^2 - cos(a)^2sin(b)^
= cos(a)^2cos(b)^2 + sin(a)^2sin(b)^2 - sin(a)^2cos(b)^2 - cos(a)^2sin(b)^
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b
= 1
Therefore, the simplified expression is 1.