Questions: Find the exact value of the expression given below. cos (π/12) cos (π/12)=

Solution Steps

To find the exact value of \(\cos\left(\frac{\pi}{12}\right)\), we can use the angle subtraction identity for cosine: \(\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)\). We can express \(\frac{\pi}{12}\) as \(\frac{\pi}{3} - \frac{\pi}{4}\). Then, we can use known values for \(\cos\) and \(\sin\) of \(\frac{\pi}{3}\) and \(\frac{\pi}{4}\) to calculate the result.

To find \(\cos\left(\frac{\pi}{12}\right)\), we can use the angle subtraction identity: \[ \cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b) \] We express \(\frac{\pi}{12}\) as \(\frac{\pi}{3} - \frac{\pi}{4}\).

We know the following trigonometric values: \[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}, \quad \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}, \quad \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}, \quad \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \]

Substituting the known values into the angle subtraction identity: \[ \cos\left(\frac{\pi}{12}\right) = \cos\left(\frac{\pi}{3}\right)\cos\left(\frac{\pi}{4}\right) + \sin\left(\frac{\pi}{3}\right)\sin\left(\frac{\pi}{4}\right) \] This gives us: \[ \cos\left(\frac{\pi}{12}\right) = \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) + \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) \]

Calculating each term: \[ \cos\left(\frac{\pi}{12}\right) = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} = \frac{\sqrt{2} + \sqrt{6}}{4} \]

Final Answer