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Question: The value of \(\sin 47^{o} + \sin 61{^\circ} - \sin 11{^\circ} - \sin 25{^\circ} =\)...

The value of sin⁡47o+sin⁡61∘−sin⁡11∘−sin⁡25∘=\sin 47^{o} + \sin 61{^\circ} - \sin 11{^\circ} - \sin 25{^\circ} =

A

sin⁡36∘\sin 36{^\circ}

B

cos⁡36∘\cos 36{^\circ}

C

sin⁡7∘\sin 7{^\circ}

D

cos⁡7∘\cos 7{^\circ}

Answer

cos⁡7∘\cos 7{^\circ}

Explanation

Solution

sin⁡47o+sin⁡61o−(sin⁡11o+sin⁡25o)\sin{}47^{o} + \sin{}61^{o} - (\sin{}11^{o} + \sin{}25^{o})

=sin⁡20osin⁡40osin⁡80ocos⁡20ocos⁡40ocos⁡80o= \frac{\sin 20^{o}\sin 40^{o}\sin 80^{o}}{\cos 20^{o}\cos 40^{o}\cos 80^{o}}

=2cos⁡7o(sin⁡54o−sin⁡18o)= 2\cos{}7^{o}(\sin{}54^{o} - \sin{}18^{o})

=2cos⁡7o.2cos⁡36o.sin⁡18o= 2\cos{}7^{o}.2\cos{}36^{o}.\sin{}18^{o}

=4.cos⁡7o.5+14.5−14=cos⁡7o.= 4.\cos{}7^{o}.\frac{\sqrt{5} + 1}{4}.\frac{\sqrt{5} - 1}{4} = \cos{}7^{o.}.