Question

Evaluate $\left(\frac{\sin 70°+\cos 40°}{\cos 70°+\sin 40°}\right)$:

• A. $1$
• B. $\frac{1}{\sqrt{3}}$
• C. $\sqrt{3}$
• D. $\frac{1}{2}$

Answer 1

The correct option is C

$\sqrt{3}$

Explanation for correct option

Simplifying the equation using trigonometric identities:

Given expression is: $\left(\frac{\sin 70°+\cos 40°}{\cos 70°+\sin 40°}\right)$

On simplifying:

$$\begin{gathered} =\left(\frac{\sin 70°+\cos 40°}{\cos 70°+\sin 40°}\right) \\ =\frac{\sin 70°+\cos \left(90°–50°\right)}{\cos \left(90°–20°\right)+\sin 40°} \\ =\frac{\sin 70°+\sin 50°}{\sin 20°+\sin 40°}\left[\because \cos \left(90°-\theta \right)=\sin \theta \right] \end{gathered}$$

Applying the formula $\sin x+\sin y=2\sin \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)$in the above equation, we get:

$$\begin{gathered} =\frac{\left[2\sin \left(\frac{70°+50°}{2}\right)\cos \left(\frac{70°-50°}{2}\right)\right]}{\left[2\sin \left(\frac{20°+40°}{2}\right)\cos \left(\frac{20°-40°}{2}\right)\right]} \\ =\frac{\sin 60°\cos 10°}{\sin 30°\cos (-10°)} \\ =\frac{\sin 60°}{\sin 30°}\because \left[\cos (-\theta )=\cos \left(\theta \right)\right] \\ =\frac{{\displaystyle \frac{\sqrt{3}}{2}}}{{\displaystyle \frac{1}{2}}} \\ =\sqrt{3} \end{gathered}$$

Therefore, the correct answer is Option (C).