If A = $60^{o}$ and B = $30^{o}$ , find the value of
$(s i n A . c o s B + c o s A . s i n B)^{2}$ + $(c o s A . c o s B - s i n A . s i n B)^{2}$ .

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Solution

Given: A = $60^{o}$ and B = $30^{o}$

$(s i n A . c o s B + c o s A . s i n B)^{2}$ + $(c o s A . c o s B - s i n A . s i n B)^{2}$

$= (s i n 60^{o} . c o s 30^{o} + c o s 60^{o} . s i n 30^{o})^{2} +$
$(c o s 60^{o} . c o s 30^{o} - s i n 60^{o} . s i n 30^{o})^{2}$

$= [(\frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{2}) + (\frac{1}{2} \times \frac{1}{2})]^{2} +$
$[(\frac{1}{2} \times \frac{\sqrt{3}}{2}) - (\frac{\sqrt{3}}{2} \times \frac{1}{2})]^{2}$

$= [(\frac{3}{4}) + (\frac{1}{4})]^{2} + [(\frac{\sqrt{3}}{4}) - (\frac{\sqrt{3}}{4})]^{2}$

$= [(\frac{3}{4}) + (\frac{1}{4})]^{2} + 0$

= $1^{2} = 1$

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