Question

If (sin θ + cos θ) = $\sqrt{3}$, then (tan θ + cot θ) = ?
(a) $\frac{1}{\sqrt{3}}$
(b) $\frac{1}{2}$
(c) 1
(d) $\frac{3}{2}$

Answer 1

(c) 1

$$\begin{gathered} (\text{sin}\theta +\text{cos}\theta )=\sqrt{3} \\ \text{=>}{(\text{sin}\theta +\text{cos}\theta )}^2=3\left[\text{Squaring both sides}\right] \\ {\text{=>sin}}^2\theta +{\text{cos}}^2\theta +2\text{sin}\theta \text{cos}\theta =3 \\ \text{=>1+}2\text{sin}\theta \text{cos}\theta =3 \\ \text{=>}2\text{sin}\theta \text{cos}\theta =2 \\ \text{=>sin}\theta \text{cos}\theta =1 \\ \therefore \text{sin}\theta \text{cos}\theta ={\text{sin}}^2\theta +{\text{cos}}^2\theta \\ \text{=>}\frac{{\text{sin}}^2\theta +{\text{cos}}^2\theta }{\text{sin}\theta \text{cos}\theta }=1 \\ \text{=>}\frac{\text{sin}\theta }{\text{cos}\theta }+\frac{\text{cos}\theta }{\text{sin}\theta }=1 \\ \text{=>tan}\theta +\text{cot}\theta =1 \end{gathered}$$