Find the Value of cos(π/12) − sin(π/2)
Question:
The value of \[ \cos\left(\frac{\pi}{12}\right)-\sin\left(\frac{\pi}{2}\right) \] is …………………………………….
The value of \[ \cos\left(\frac{\pi}{12}\right)-\sin\left(\frac{\pi}{2}\right) \] is …………………………………….
Solution
We know that
\[ \sin\left(\frac{\pi}{2}\right)=1 \]
Also,
\[ \cos\left(\frac{\pi}{12}\right) = \cos15^\circ \]
Using the identity:
\[ \cos(A-B)=\cos A\cos B+\sin A\sin B \]
Take
\[ A=45^\circ, \qquad B=30^\circ \]
Then,
\[ \cos15^\circ = \cos45^\circ\cos30^\circ + \sin45^\circ\sin30^\circ \]
\[ = \frac{1}{\sqrt2}\cdot\frac{\sqrt3}{2} + \frac{1}{\sqrt2}\cdot\frac12 \]
\[ = \frac{\sqrt3+1}{2\sqrt2} \]
Therefore,
\[ \cos\left(\frac{\pi}{12}\right)-\sin\left(\frac{\pi}{2}\right) = \frac{\sqrt3+1}{2\sqrt2}-1 \]
Rationalizing,
\[ = \frac{\sqrt6+\sqrt2-4}{4} \]
Therefore,
\[ \boxed{ \frac{\sqrt6+\sqrt2-4}{4} } \]