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Question

The value of $sin invisible function application 47 to the power of ring operator plus sin invisible function application 61 to the power of ring operator minus sin invisible function application 11 to the power of ring operator minus sin invisible function application 25 to the power of ring operator$ is

$\sin 7^{0}$
$\cos 7^{\circ}$
$\sin 14^{\circ}$
$\cos 14^{\circ}$

hint Hint:

Using sin and cosine trigonometry formulae, find the answer.

The correct answer is: $cos invisible function application 7 to the power of ring operator$

$sin 47 to the power of ring operator plus sin 61 to the power of ring operator minus sin 11 to the power of ring operator minus sin 25 to the power of ring operator equals open parentheses sin 47 to the power of ring operator minus sin 25 close parentheses to the power of ring operator plus open parentheses sin 61 to the power of ring operator minus sin 11 to the power of ring operator close parentheses U s i n g space f o r m u l a space sin A minus sin B equals 2 cos open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses sin open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses comma equals 2 cos open parentheses fraction numerator 47 degree plus 25 degree over denominator 2 end fraction close parentheses sin open parentheses fraction numerator 47 degree minus 25 degree over denominator 2 end fraction close parentheses plus 2 cos open parentheses fraction numerator 61 degree plus 11 degree over denominator 2 end fraction close parentheses sin open parentheses fraction numerator 61 degree minus 11 degree over denominator 2 end fraction close parentheses equals 2 cos open parentheses fraction numerator 72 degree over denominator 2 end fraction close parentheses sin open parentheses fraction numerator 22 degree over denominator 2 end fraction close parentheses plus 2 cos open parentheses fraction numerator 72 degree over denominator 2 end fraction close parentheses sin open parentheses fraction numerator 50 degree over denominator 2 end fraction close parentheses equals 2 cos 36 degree sin 11 degree plus 2 cos 36 degree sin 25 degree equals 2 cos 36 degree open parentheses sin 11 degree plus sin 25 degree close parentheses U s i n g space f o r m u l a space sin A plus sin B equals 2 sin open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses cos open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses comma equals 2 cos 36 degree open parentheses sin 11 degree plus sin 25 degree close parentheses equals 2 cos 36 degree open parentheses 2 sin open parentheses fraction numerator 11 degree plus 25 degree over denominator 2 end fraction close parentheses cos open parentheses fraction numerator 11 degree minus 25 degree over denominator 2 end fraction close parentheses close parentheses equals 2 cos 36 degree open parentheses 2 sin open parentheses fraction numerator 36 degree over denominator 2 end fraction close parentheses cos open parentheses fraction numerator negative 14 degree over denominator 2 end fraction close parentheses close parentheses equals 2 cos 36 degree open parentheses 2 sin open parentheses 18 degree close parentheses cos open parentheses negative 7 degree close parentheses close parentheses equals 4 cos 36 degree sin open parentheses 18 degree close parentheses cos open parentheses 7 degree close parentheses space space space space space space space space space space space space space space space space space space space space space space open parentheses because cos open parentheses negative theta close parentheses equals cos theta close parentheses W e space k n o w space t h a t space sin 18 degree equals open parentheses fraction numerator square root of 5 minus 1 over denominator 4 end fraction close parentheses space a n d space cos 36 degree equals open parentheses fraction numerator square root of 5 plus 1 over denominator 4 end fraction close parentheses therefore 4 cross times open parentheses fraction numerator square root of 5 plus 1 over denominator 4 end fraction close parentheses cross times open parentheses fraction numerator square root of 5 minus 1 over denominator 4 end fraction close parentheses cross times cos open parentheses 7 degree close parentheses equals open parentheses fraction numerator 5 minus 1 over denominator 4 end fraction close parentheses cross times cos open parentheses 7 degree close parentheses equals cos 7 degree$

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