If tan α = x + 1, tanβ = x - 1, show that 2 cot(α - β) = x².

Prove that:

If sin α sin β - cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0.

IfAngle θ is divided into two parts such that the tangents of the one part is λ times the tangent of other, And ϕ is their difference, then show that .

If , then show that sin α + cos α = cos x.