Find the Value of cos²6° – cos²24°
Question:
\[ \cos^2 6^\circ-\cos^2 24^\circ \]Solution
Use the identity
\[ \cos^2 A-\cos^2 B = (\cos A-\cos B)(\cos A+\cos B) \]Also,
\[ \cos C-\cos D = -2\sin\frac{C+D}{2}\sin\frac{C-D}{2} \] \[ \cos C+\cos D = 2\cos\frac{C+D}{2}\cos\frac{C-D}{2} \]Therefore,
\[ \cos^2 A-\cos^2 B = -\sin(C+D)\sin(C-D) \]Taking \(A=6^\circ\) and \(B=24^\circ\),
\[ \cos^2 6^\circ-\cos^2 24^\circ = -\sin30^\circ\sin(-18^\circ) \] \[ = -\frac12(-\sin18^\circ) \] \[ = \frac12\sin18^\circ \]Using the standard value
\[ \sin18^\circ = \frac{\sqrt5-1}{4} \]Hence,
\[ \cos^2 6^\circ-\cos^2 24^\circ = \frac12\cdot\frac{\sqrt5-1}{4} \] \[ = \frac{\sqrt5-1}{8} \]