MCQ
If $\cos\text{A}+\cos^2\text{A}=1$ then $\big(\sin^2\text{A}+\sin^4\text{A}\big)=?$
- A$\frac{1}2{}$
- B$2$
- ✓$1$
- D$4$
✓
Answer
Correct option: C.
$1$
$\cos\text{A}+\cos^2\text{A}=1$
$\Rightarrow\cos\text{A}=\sin^2\text{A}\dots(\text{i})$
Squaring both sides of (i), we get:
$\cos^2\text{A}=\sin^4\text{A}\dots(\text{ii})$
Adding (i) and (ii), we get:
$\sin^2\text{A}=\sin^4\text{A}=\cos\text{A}+\cos^2\text{A}$
$\Rightarrow\sin^2\text{A}+\sin^4\text{A}=1\ [\because\cos\text{A}+\cos^2\text{A}=1]$
$\Rightarrow\cos\text{A}=\sin^2\text{A}\dots(\text{i})$
Squaring both sides of (i), we get:
$\cos^2\text{A}=\sin^4\text{A}\dots(\text{ii})$
Adding (i) and (ii), we get:
$\sin^2\text{A}=\sin^4\text{A}=\cos\text{A}+\cos^2\text{A}$
$\Rightarrow\sin^2\text{A}+\sin^4\text{A}=1\ [\because\cos\text{A}+\cos^2\text{A}=1]$
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