Question
Prove the following trigonometric identities.
If $\cos\text{A}+\cos^2\text{A}=1,$ prove that $\sin^2\text{A}+\sin^4\text{A}=1.$
If $\cos\text{A}+\cos^2\text{A}=1,$ prove that $\sin^2\text{A}+\sin^4\text{A}=1.$
✓
Answer
$\cos\text{A}+\cos^2\text{A}=1$
$\cos\text{A}=1-\cos^2\text{A}$
$\cos\text{A}=\sin^2\text{A}$
$\text{L.H.S}=\sin^2\text{A}+\sin^4\text{A}$
$=\sin^2\text{A}+(\sin^2\text{A})^2$
$=\sin^2\text{A}+(\cos\text{A})^2$
$=\sin^2\text{A}+\cos^2\text{A}$
$=1=\text{R.H.S}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$
$\cos\text{A}=1-\cos^2\text{A}$
$\cos\text{A}=\sin^2\text{A}$
$\text{L.H.S}=\sin^2\text{A}+\sin^4\text{A}$
$=\sin^2\text{A}+(\sin^2\text{A})^2$
$=\sin^2\text{A}+(\cos\text{A})^2$
$=\sin^2\text{A}+\cos^2\text{A}$
$=1=\text{R.H.S}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$
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