Find the value of \sin ^{4} 30^{\circ}+\cos ^{4} 30^{\circ}-\sin 25^{\circ} \cos 65^{\circ} -\sin 65^{\circ} \cos 25^{\circ}

SSC CGL 18 Aug 2021 Shift 2 Paper

Question : 72 of 100

Marks: +1, -0

Find the value of

\sin ^{4} 30^{∘}+\cos ^{4} 30^{∘}-\sin 25^{∘} \cos 65^{∘}

-\sin 65^{∘} \cos 25^{∘}

Solution:

\sin ^{4} 30^{∘}+\cos ^{4} 30^{∘}

\;\; -\sin 25^{∘} \cos 65^{∘}-\sin 65^{∘} \cos 25^{∘}

=\sin ^{4} 30^{∘}+\cos ^{4} 30^{∘}-\sin (65^{∘}+25^{∘})

\;\; [ ∵ \sin (A+B)=\sin A \cos B

\;\; +\cos A \sin B]

=\sin ^{4} 30^{∘}+\cos ^{4} 30^{∘}-\sin 90^{∘}

=(\;{1}/{2})^{4}+(\;{√{3}}/{2})^{4}-1=\;{1}/{16}+\;{9}/{16}-1

=\;{10}/{16}-1=\;{5}/{8}-1=\;{5-8}/{8}=-\;{3}/{8}

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