Let a line makes an angle θ with y-,z-axes and ϕ  with x-axis. If  sinϕ and sinθare in the ratiom:n , then cosθ=

Given  ϕ,θ,θare the angles made by a line with axes, it implies that  cos2ϕ+cos2θ+cos2θ=12cos2θ=1−cos2ϕ=sin2ϕ           Given  sinϕsinθ=mn⇒sinϕ=mnsinθSubstitute  sinϕ=mnsinθ in the equation  2cos2θ=sin2ϕIt implies that  2cos2θ=m2n2sin2θtan2θ=2n2m2Use right triangle, we get  cosθ=m2n2+m2