Question
Simplest form of $(1-\cos^2\text{A})(1+\cos^2\text{A})$ is __________.
✓
Answer
Simplest form of $(1-\cos^2\text{A})(1+\cos^2\text{A})$ is 1.Solution:
We know that
$1-\cos^2\text{A}=\sin^2\text{A}$
$1-\cot^2\text{A}=\text{cosec}^2\text{ A}$
$=(1-\cos^2\text{A})(1+\cos^2\text{A})$
$=\sin^2\text{A}.\text{cosec}^2\text{ A}$
$=\sin^2\text{A}.\frac{1}{\sin^2\text{A}}[\text{cosec}^2\text{ A}=\frac{1}{\sin^2\text{A}}]$
$=1$
We know that
$1-\cos^2\text{A}=\sin^2\text{A}$
$1-\cot^2\text{A}=\text{cosec}^2\text{ A}$
$=(1-\cos^2\text{A})(1+\cos^2\text{A})$
$=\sin^2\text{A}.\text{cosec}^2\text{ A}$
$=\sin^2\text{A}.\frac{1}{\sin^2\text{A}}[\text{cosec}^2\text{ A}=\frac{1}{\sin^2\text{A}}]$
$=1$
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