If sin θ+ cos θ=√ 3, then prove that tan θ+ cot θ=1

\[\begin{aligned} & (\sin \theta+\cos \theta)^{2}=(\sqrt{3})^{2} \\ & \Rightarrow \sin ^{2} \theta+\cos ^{2} \theta+2 \sin \theta \cos \theta=3 \\ & \Rightarrow \sin \theta \cos \theta=1 \quad--(\mathrm{i}) \\ & \text { LHS }=\tan \theta+\cot \theta=\frac{\sin \theta}{\cos \theta}+\frac{\cos \theta}{\sin \theta}=\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\cos \theta \sin \theta}=\frac{1}{\cos \theta \sin \theta} \\ & \quad=1 \quad[\text { using (i)] } \\ & \quad=\text { RHS } \end{aligned} \]