Q77 / 180 · COMEDK 2024 (Afternoon Shift) · Mathematics
$\left(\cos \dfrac{\pi}{12}-\sin \dfrac{\pi}{12}\right)\left(\tan \dfrac{\pi}{12}+\cot \dfrac{\pi}{12}\right)=$
- A$\sqrt{2}$
- Correct answerB$2 \sqrt{2}$
- C$4 \sqrt{2}$
- D$\dfrac{1}{\sqrt{2}}$
Solution
Let $x = \dfrac{\pi}{12}$. The expression is $(\cos x - \sin x)(\tan x + \cot x)$.
We know that $\tan x + \cot x = \dfrac{\sin x}{\cos x} + \dfrac{\cos x}{\sin x} = \dfrac{\sin^2 x + \cos^2 x}{\sin x \cos x} = \dfrac{1}{\sin x \cos x}$.
Multiplying by $2$ in the numerator and denominator, we get $\dfrac{2}{2 \sin x \cos x} = \dfrac{2}{\sin 2x}$.
Substituting $x = \dfrac{\pi}{12}$, $2x = \dfrac{\pi}{6}$. Thus, $\tan x + \cot x = \dfrac{2}{\sin(\pi/6)} = \dfrac{2}{1/2} = 4$.
Now consider the first term: $\cos x - \sin x$.
Using the identity $\cos x - \sin x = \sqrt{2} \left( \dfrac{1}{\sqrt{2}} \cos x - \dfrac{1}{\sqrt{2}} \sin x \right) = \sqrt{2} \cos(x + \pi/4)$.
For $x = \pi/12$, $x + \pi/4 = \pi/12 + 3\pi/12 = 4\pi/12 = \pi/3$.
So, $\cos(\pi/12) - \sin(\pi/12) = \sqrt{2} \cos(\pi/3) = \sqrt{2} \times \dfrac{1}{2} = \dfrac{1}{\sqrt{2}}$.
The product is $\dfrac{1}{\sqrt{2}} \times 4 = \dfrac{4}{\sqrt{2}} = 2\sqrt{2}$.
Answer: $2 \sqrt{2}$
We know that $\tan x + \cot x = \dfrac{\sin x}{\cos x} + \dfrac{\cos x}{\sin x} = \dfrac{\sin^2 x + \cos^2 x}{\sin x \cos x} = \dfrac{1}{\sin x \cos x}$.
Multiplying by $2$ in the numerator and denominator, we get $\dfrac{2}{2 \sin x \cos x} = \dfrac{2}{\sin 2x}$.
Substituting $x = \dfrac{\pi}{12}$, $2x = \dfrac{\pi}{6}$. Thus, $\tan x + \cot x = \dfrac{2}{\sin(\pi/6)} = \dfrac{2}{1/2} = 4$.
Now consider the first term: $\cos x - \sin x$.
Using the identity $\cos x - \sin x = \sqrt{2} \left( \dfrac{1}{\sqrt{2}} \cos x - \dfrac{1}{\sqrt{2}} \sin x \right) = \sqrt{2} \cos(x + \pi/4)$.
For $x = \pi/12$, $x + \pi/4 = \pi/12 + 3\pi/12 = 4\pi/12 = \pi/3$.
So, $\cos(\pi/12) - \sin(\pi/12) = \sqrt{2} \cos(\pi/3) = \sqrt{2} \times \dfrac{1}{2} = \dfrac{1}{\sqrt{2}}$.
The product is $\dfrac{1}{\sqrt{2}} \times 4 = \dfrac{4}{\sqrt{2}} = 2\sqrt{2}$.
Answer: $2 \sqrt{2}$
Hence, the correct answer is option: 'B'.