Answer:
=o
Explanation:
The expression cot(25)tan(65) - tan(35)cot(55) simplifies to (cos(25)/sin(25))(sin(65)/cos(65)) - (sin(35)/cos(35))(cos(55)/sin(55)).
Multiplying the numerators and denominators of each fraction to obtain a common denominator, we have:
cos(25)sin(65)cos(35)sin(55)/[sin(25)cos(65)cos(35)sin(55)] - sin(35)cos(55)sin(25)cos(65)/[cos(35)sin(55)sin(25)cos(65)]
The two fractions have the same denominator, so we can combine the numerators:
cos(25)sin(65)cos(35)sin(55) - sin(35)cos(55)sin(25)cos(65)
Using the identity sin(90 - x) = cos(x), we can rewrite sin(65) as cos(25) and sin(55) as cos(35):
cos(25)cos(25)cos(35)cos(25) - sin(35)cos(55)sin(25)cos(65)
Using the identity sin(2x) = 2sin(x)cos(x), we can rewrite sin(25)cos(65) as (1/2)sin(50):
cos(25)cos(25)cos(35)cos(25) - (1/2)sin(50)sin(70)
Using the identity sin(2x) = 2sin(x)cos(x), we can rewrite sin(50) as 2sin(25)cos(25):
cos(25)cos(25)cos(35)cos(25) - sin(25)cos(25)cos(70)
Using the identity cos(90 - x) = sin(x), we can rewrite cos(70) as sin(20):
cos(25)cos(25)cos(35)cos(25) - sin(25)cos(25)sin(20)
Using the identity sin(2x) = 2sin(x)cos(x), we can rewrite sin(20) as (1/2)sin(40):
cos(25)cos(25)cos(35)cos(25) - (1/2)sin(40)cos(25)