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Evaluate each of the following:$ \sin ^{2} 30^{\circ} \cos ^{2} 45^{\circ}+4 \tan ^{2} 30^{\circ}+\frac{1}{2} \sin ^{2} 90^{\circ}-2 \cos ^{2} 90^{\circ}+\frac{1}{24} \cos ^{2} 0^{\circ} $
Given:
\( \sin ^{2} 30^{\circ} \cos ^{2} 45^{\circ}+4 \tan ^{2} 30^{\circ}+\frac{1}{2} \sin ^{2} 90^{\circ}-2 \cos ^{2} 90^{\circ}+\frac{1}{24} \cos ^{2} 0^{\circ} \)
To do:
We have to evaluate \( \sin ^{2} 30^{\circ} \cos ^{2} 45^{\circ}+4 \tan ^{2} 30^{\circ}+\frac{1}{2} \sin ^{2} 90^{\circ}-2 \cos ^{2} 90^{\circ}+\frac{1}{24} \cos ^{2} 0^{\circ} \).
Solution:
We know that,
$\sin 30^{\circ}=\frac{1}{2}$
$\cos 45^{\circ}=\frac{1}{\sqrt2}$
$\tan 30^{\circ}=\frac{1}{\sqrt3}$
$\sin 90^{\circ}=1$
$\cos 90^{\circ}=0$
$\cos 0^{\circ}=1$
$\sin ^{2} 30^{\circ} \cos ^{2} 45^{\circ}+4 \tan ^{2} 30^{\circ}+\frac{1}{2} \sin ^{2} 90^{\circ}-2 \cos ^{2} 90^{\circ}+\frac{1}{24} \cos ^{2} 0^{\circ}=\left(\frac{1}{2}\right)^{2} \times \left(\frac{1}{\sqrt{2}}\right)^{2} +4\left(\frac{1}{\sqrt{3}}\right)^{2} +\frac{1}{2} \times ( 1)^{2} -2( 0)^{2} +\frac{1}{24}( 1)^{2}$
$=\frac{1}{4} \times \frac{1}{2} +4\times \frac{1}{3} +\frac{1}{2} -0+\frac{1}{24}$
$=\frac{1}{8} +\frac{4}{3} +\frac{1}{2} +\frac{1}{24}$
$=\frac{1( 3) +4( 8) +1( 12) +1}{24}$
$=\frac{3+32+12+1}{24}$
$=\frac{48}{24}$
$=2$
Hence, $\sin ^{2} 30^{\circ} \cos ^{2} 45^{\circ}+4 \tan ^{2} 30^{\circ}+\frac{1}{2} \sin ^{2} 90^{\circ}-2 \cos ^{2} 90^{\circ}+\frac{1}{24} \cos ^{2} 0^{\circ}=2$.
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