Question
If $A = 30^\circ ;$ show that:$ \cos 2A = \cos^4A - \sin^4A$
✓
Answer
Given that $A = 30^\circ$
$\text{LHS} = \cos 2A$
$= \cos 2(30^\circ )$
$= \cos 60^\circ$
$=\frac{1}{2}$
$\text { RHS }=\cos ^4 A -\sin ^4 A$
$=\cos ^4 30^{\circ}-\sin ^4 30^{\circ}$
$=\left(\frac{\sqrt{3}}{2}\right)^4-\left(\frac{1}{2}\right)^4$
$=\frac{9}{16}-\frac{1}{16}$
$=\frac{1}{2}$
$\text{LHS}=\text{RHS}$
$\text{LHS} = \cos 2A$
$= \cos 2(30^\circ )$
$= \cos 60^\circ$
$=\frac{1}{2}$
$\text { RHS }=\cos ^4 A -\sin ^4 A$
$=\cos ^4 30^{\circ}-\sin ^4 30^{\circ}$
$=\left(\frac{\sqrt{3}}{2}\right)^4-\left(\frac{1}{2}\right)^4$
$=\frac{9}{16}-\frac{1}{16}$
$=\frac{1}{2}$
$\text{LHS}=\text{RHS}$
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