Question

If sin A + cos A = \[\sqrt3\]; prove that: tan A + cot A = 1

Answer 1

Given,

⇒ sin A + cos A = \[\sqrt3\]

Squaring both sides:

⇒ (sin A + cos A)² = \[{(\sqrt3)}^2\]

⇒ sin² A + cos² A + 2 sin A cos A = 3

By formula,

sin² A + cos² A = 1

⇒ 1 + 2 sin A cos A = 3

⇒ 2 sin A cos A = 2

⇒ sin A cos A = 1

Now, solving tan A + cot A:

⇒ tan A + cot A

\[\Rightarrow \frac{\sin A}{\cos A} + \frac{\cos A}{\sin A}\]

\[\Rightarrow \frac{\sin^2 A + \cos^2 A}{\sin A \cos A}\]

\[\Rightarrow \frac{1}{\sin A \cos A}\]

\[\Rightarrow \frac{1}{1}\]

⇒ 1

Since, L.H.S. = R.H.S.