Question

Q. If sin $\theta$ + cos $\theta$= $\sqrt{2}$ , then evaluate : tan $\theta$ + cot $\theta$

Answer 1

$$\begin{gathered} \sin \mathrm{\theta }+\cos \mathrm{\theta }=\sqrt{2} \\ \Rightarrow {\sin }^2\mathrm{\theta }+{\cos }^2\mathrm{\theta }+2\sin \mathrm{\theta }.\cos \mathrm{\theta }=2 \\ \Rightarrow 1+2\sin \mathrm{\theta }.\cos \mathrm{\theta }=2 \\ \Rightarrow 2\sin \mathrm{\theta }.\cos \mathrm{\theta }=1 \\ \Rightarrow \sin \mathrm{\theta }.\cos \mathrm{\theta }=\frac{1}{2} \\ \mathrm{Now},\tan \mathrm{\theta }+\cot \mathrm{\theta } \\ =\frac{\sin \mathrm{\theta }}{\cos \mathrm{\theta }}+\frac{\cos \mathrm{\theta }}{\sin \mathrm{\theta }} \\ =\frac{{\sin }^2\mathrm{\theta }+{\cos }^2\mathrm{\theta }}{\sin \mathrm{\theta }.\cos \mathrm{\theta }} \\ =\frac{1}{1/2} \\ =2 \end{gathered}$$