MCQ
If $\cos\text{A}+\cos^2\text{A}=1,$ then $\sin^2\text{A}+\sin^4\text{A}=$
- A$-1$
- B$0$
- ✓$1$
- DNone of these.
✓
Answer
Correct option: C.
$1$
Given,
$\cos\text{A}+\cos^2\text{A}=1$
$\Rightarrow\ 1-\cos^2\text{A}=\cos\text{A}$
So,
$\sin^2\text{A}+\sin^4\text{A}$
$=\sin^2\text{A}+\sin^2\text{A}\sin^2\text{A}$
$=\sin^2\text{A}+(1-\cos^2\text{A})(1-\cos^2\text{A})$
$=\sin^2\text{A}+\cos\text{A}\cos\text{A}$
$=\sin^2\text{A}+\cos^2\text{A}$
$=1$
Hence, the correct option is $(c).$
$\cos\text{A}+\cos^2\text{A}=1$
$\Rightarrow\ 1-\cos^2\text{A}=\cos\text{A}$
So,
$\sin^2\text{A}+\sin^4\text{A}$
$=\sin^2\text{A}+\sin^2\text{A}\sin^2\text{A}$
$=\sin^2\text{A}+(1-\cos^2\text{A})(1-\cos^2\text{A})$
$=\sin^2\text{A}+\cos\text{A}\cos\text{A}$
$=\sin^2\text{A}+\cos^2\text{A}$
$=1$
Hence, the correct option is $(c).$
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