If $\sin heta + \cos heta = \sqrt{2}$, prove that $ an heta + \cot heta = 2$.

Question

If $\sin 	heta + \cos 	heta = \sqrt{2}$, prove that $	an 	heta + \cot 	heta = 2$.

Accepted Answer

Step 1: $\sin 	heta + \cos 	heta = \sqrt{2}$. Squaring both sides: $\sin^2 	heta + \cos^2 	heta + 2\sin 	heta \cos 	heta = 2$
Step 2: $1 + 2\sin 	heta \cos 	heta = 2 \Rightarrow \sin 	heta \cos 	heta = \frac{1}{2}$. So $	an 	heta + \cot 	heta = \frac{\sin 	heta}{\cos 	heta} + \frac{\cos 	heta}{\sin 	heta}$
Step 3: $= \frac{\sin^2 	heta + \cos^2 	heta}{\sin 	heta \cos 	heta} = \frac{1}{\frac{1}{2}} = 2$. Hence $	an 	heta + \cot 	heta = 2$

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