Question

The value of $\frac{\sin \left(55°\right)-\cos \left(55°\right)}{\sin \left(10°\right)}$ is

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

Answer 1

The correct option is D

$\sqrt{2}$

The explanation for the correct option

The given trigonometric expression: $\frac{\sin \left(55°\right)-\cos \left(55°\right)}{\sin \left(10°\right)}$.

$$\begin{gathered} \frac{\sin \left(55°\right)-\cos \left(55°\right)}{\sin \left(10°\right)}=\frac{\sin \left(55°\right)-\cos \left(90°-35°\right)}{\sin \left(10°\right)} \\ \Rightarrow \frac{\sin \left(55°\right)-\cos \left(55°\right)}{\sin \left(10°\right)}=\frac{\sin \left(55°\right)-\sin \left(35°\right)}{\sin \left(10°\right)}\left[\because \cos \left(90°-\theta \right)=\sin \left(\theta \right)\right] \\ \Rightarrow \frac{\sin \left(55°\right)-\cos \left(55°\right)}{\sin \left(10°\right)}=\frac{2\cos \left({\displaystyle \frac{55°+35°}{2}}\right)\sin \left({\displaystyle \frac{55°-35°}{2}}\right)}{\sin \left(10°\right)}\mathbf{[}\mathbf{\because }\mathbf{sin}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{-}\mathbf{sin}\mathbf{\left(}\mathbf{B}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{cos}\mathbf{\left(}\frac{\mathbf{A}\mathbf{+}\mathbf{B}}{\mathbf{2}}\mathbf{\right)}\mathbf{sin}\mathbf{\left(}\frac{\mathbf{A}\mathbf{-}\mathbf{B}}{\mathbf{2}}\mathbf{\right)}\mathbf{]} \\ \Rightarrow \frac{\sin \left(55°\right)-\cos \left(55°\right)}{\sin \left(10°\right)}=\frac{2\cos \left({\displaystyle \frac{90°}{2}}\right)\sin \left({\displaystyle \frac{20°}{2}}\right)}{\sin \left(10°\right)} \\ \Rightarrow \frac{\sin \left(55°\right)-\cos \left(55°\right)}{\sin \left(10°\right)}=\frac{2\cos \left({\displaystyle 45°}\right)\sin \left({\displaystyle 10°}\right)}{\sin \left(10°\right)} \\ \Rightarrow \frac{\sin \left(55°\right)-\cos \left(55°\right)}{\sin \left(10°\right)}=2\times \frac{1}{\sqrt{2}} \\ \Rightarrow \frac{\sin \left(55°\right)-\cos \left(55°\right)}{\sin \left(10°\right)}=\sqrt{2} \end{gathered}$$

Therefore, the value of $\frac{\sin \left(55°\right)-\cos \left(55°\right)}{\sin \left(10°\right)}$ is $\sqrt{2}$.

Hence, the correct option is (D).