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Which one of the following is correct? $\left(1+\cos 67 \frac{1^{\circ}}{2}\right)\left(1+\cos 112 \frac{1^{\circ}}{2}\right)$ is
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The correct answer is:
an irrational number and is less than $1$
The given expression $\left(1+\cos 67 \frac{1^{\circ}}{2}\right)\left(1+\cos 112 \frac{1^{\circ}}{2}\right)$
Can also be writters as :
$\left(1+\cos 67 \frac{1^{\circ}}{2}\right)\left\{1+\cos \left(180^{\circ}-67 \frac{1^{\circ}}{2}\right)\right\}$
$=\left(1+\cos 67 \frac{1^{\circ}}{2}\right)\left(1-\cos 67 \frac{1^{\circ}}{2}\right)$
$=1-\cos ^{2} 67^{\circ} \frac{1^{\circ}}{2}=\sin ^{2} 67 \frac{1^{\circ}}{2}$
$=\frac{1-\cos 135^{\circ}}{2}=\frac{\sqrt{2}+1}{2 \sqrt{2}} \quad\left(\because \sin ^{2} \mathrm{~A}=\frac{1-\cos 2 \mathrm{~A}}{2}\right)$
Which is an irrational number and is less than 1
Can also be writters as :
$\left(1+\cos 67 \frac{1^{\circ}}{2}\right)\left\{1+\cos \left(180^{\circ}-67 \frac{1^{\circ}}{2}\right)\right\}$
$=\left(1+\cos 67 \frac{1^{\circ}}{2}\right)\left(1-\cos 67 \frac{1^{\circ}}{2}\right)$
$=1-\cos ^{2} 67^{\circ} \frac{1^{\circ}}{2}=\sin ^{2} 67 \frac{1^{\circ}}{2}$
$=\frac{1-\cos 135^{\circ}}{2}=\frac{\sqrt{2}+1}{2 \sqrt{2}} \quad\left(\because \sin ^{2} \mathrm{~A}=\frac{1-\cos 2 \mathrm{~A}}{2}\right)$
Which is an irrational number and is less than 1
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