tanθ + cotθ = 1/sinθcosθ
since we know that;
tanθ = sinθ/cosθ, and
cotθ = cosθ/sinθ
now when we add tanθ and cotθ and replace their values;
tanθ + cotθ=sinθ/cosθ + cosθ/sinθ
For a common denominator to add those two fractions, the obvious choice is sinθ.cosθ , so
tanθ + cotθ = sin²θ/sinθcosθ + cos²θ/sinθcosθ =sin²θ + cos²θ / sinθcosθ
now we can use the identity that;
sin²θ + cos²θ = 1
So,
tanθ + cotθ = 1/sinθcosθ