1) cos(a)cos(3a) - sin(a)sin(3a)
= cos(a)(4cos(a)^3 - 3cos(a)) - sin(a)(3sin(a) - 4sin(a)^3)
= 4cos(a)^4 - 3cos(a)^2 - 3sin(a)^2 + 4sin(a)^4
= 4*cos(a)^4 - 3

2) sin(2a)cos(a) + cos(2a)sin(a)
= 2sin(a)cos(a)cos(a) + (2cos(a)^2 - 1)sin(a)
= 2sin(a)cos(a)^2 + 2sin(a)cos(a)^2 - sin(a)
= 4sin(a)*cos(a)^2 - sin(a)

3) sin(40)cos(5) + cos(40)sin(5)
= sin(40+5)
= sin(45)
= 1/√2

4) (tan(7π/15) - tan(2π/15)) / (1 + tan(7π/15)tan(2π/15)
= (tan(1/3π) - tan(2π/15)) / (1 + tan(1/3π)tan(2π/15))
= (tan(1/3π) - tan(1/3π)) / (1 + tan(1/3π)*tan(1/3π))
= 0/1
= 0

5) sin(a) = 4/5, where a is an angle in radians
There can be multiple values of a that satisfy this equation. One possible angle is sin⁻¹(4/5) = 0.9273 radians.