sin21cos9 - cos84cos6 = 1/4

Dear student

Consider, \sin 21 ° \cos 9 ° = \frac{1}{2} [\sin (21 ° + 9 °) + \sin (21 ° - 9 °)] [∵ 2 \sin A \cos B = \sin (A + B) + \sin (A - B)] \Rightarrow \sin 21 ° \cos 9 ° = \frac{1}{2} [\sin 30 ° + \sin 12 °] . . . . . . . . (1) and \cos 84 ° \cos 6 ° = \sin (90 ° - 84 °) \cos 6 ° \Rightarrow \cos 84 ° \cos 6 ° = \sin 6 ° \cos 6 ° . . . . . . . . . . . (2) Therefore, \sin 21 ° \cos 9 ° - \cos 84 ° \cos 6 ° = \frac{1}{2} [\sin 30 ° + \sin 12 °] - \sin 6 ° \cos 6 ° [from 1 and 2] \Rightarrow \sin 21 ° \cos 9 ° - \cos 84 ° \cos 6 ° = \frac{1}{2} [\frac{1}{2} + \sin 12 °] - \frac{1}{2} \sin 12 ° [\sin 2 A = 2 \sin Acos A] \Rightarrow \sin 21 ° \cos 9 ° - \cos 84 ° \cos 6 ° = \frac{1}{4} + \frac{1}{2} \sin 12 ° - \frac{1}{2} \sin 12 ° = \frac{1}{4}

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