If sin θ + cos θ = √3, then what is tan θ + cot θ equal to?

Explanation -

Let's square the given equation \( \sin \theta + \cos \theta = \sqrt{3}\)  to make use of trigonometric identities:

\( (\sin \theta + \cos \theta)^2 = (\sqrt{3})^2\)
\( \sin^2 \theta + 2 \sin \theta \cos \theta + \cos^2 \theta = 3\)

Using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) :

\( 1 + 2 \sin \theta \cos \theta = 3 \\ 2 \sin \theta \cos \theta = 2 \\ \sin \theta \cos \theta = 1 \)

we know that -

\(\tan \theta + \cot \theta = \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} \)

Using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\)  and \( \sin \theta \cos \theta = 1 :\)

\( \tan \theta + \cot \theta = \frac{1}{1} = 1 \)

Therefore, \( \tan \theta + \cot \theta = 1 .\)