Question
Evaluate the following:
If A = 45°, verify that:
$\cos2\text{A}=2\cos^2\text{A}-1=1-2\sin^2\text{A}$
If A = 45°, verify that:
$\cos2\text{A}=2\cos^2\text{A}-1=1-2\sin^2\text{A}$
✓
Answer
$\text{A}=45^\circ$
$\Rightarrow2\text{A}=2\times45^\circ=90^\circ$
$\cos2\text{A}=2\cos90^\circ=0$
$2\cos^2\text{A}-1=2\cos^245-1$
$=2\times\Big(\frac{1}{\sqrt{2}}\Big)^2-1$
$=2\times\frac12-1=1-1=0$
Now, $1-2\sin^2\text{A}=1-2\times\Big(\frac{1}{\sqrt{2}}\Big)^2$
$=1-2\times\frac12=1-1=0$
$\therefore\ \cos2\text{A}=2\cos^2\text{A}-1=1-2\sin^2\text{A}$
$\Rightarrow2\text{A}=2\times45^\circ=90^\circ$
$\cos2\text{A}=2\cos90^\circ=0$
$2\cos^2\text{A}-1=2\cos^245-1$
$=2\times\Big(\frac{1}{\sqrt{2}}\Big)^2-1$
$=2\times\frac12-1=1-1=0$
Now, $1-2\sin^2\text{A}=1-2\times\Big(\frac{1}{\sqrt{2}}\Big)^2$
$=1-2\times\frac12=1-1=0$
$\therefore\ \cos2\text{A}=2\cos^2\text{A}-1=1-2\sin^2\text{A}$
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