|
|
| Line 3: |
Line 3: |
| <math>\frac{\text{A}^2}{\left( 1+\text{A}^2\right)^2} + \frac{\text{B}^2}{\left(1+\text{B}^2\right)^2} = \frac{1}{4}</math>. | | <math>\frac{\text{A}^2}{\left( 1+\text{A}^2\right)^2} + \frac{\text{B}^2}{\left(1+\text{B}^2\right)^2} = \frac{1}{4}</math>. |
|
| |
|
|
| |
| == Solution ==
| |
| <cmath>\begin{align*}
| |
| \frac{\text{A}}{1+\text{A}^2} &= \frac{\frac{1-\cos\theta}{\sin\theta}}{1+\left(\frac{1- \cos\theta}{\sin\theta}\right)^2} \\
| |
| &= \frac{\frac{1-\cos\theta}{\sin\theta}}{\frac{\sin^2\theta+ \cos^2\theta-2\cos\theta+1}{\sin^2\theta}} \\
| |
| &= \frac{\frac{1-\cos\theta}{\sin\theta}}{\frac{2\left(1-\cos\theta\right)}{\sin^2\theta}} \\
| |
| &= \frac{\sin\theta}{2}
| |
| \end{align*}</cmath>
| |
|
| |
| Similarly <math>\frac{\text{B}}{1+\text{B}^2} = \frac{\cos\theta}{2}</math>
| |
|
| |
| So <cmath>\begin{align*}
| |
| \frac{\text{A}^2}{\left(1+\text{A}^2\right)^2} + \frac{\text{B}^2}{\left(1+\text{B}^2\right)^2} &= \frac{\sin^2\theta}{2^2} + \frac{\cos^2\theta}{2^2} \\
| |
| &= \frac{1}{4}
| |
| \end{align*}</cmath>
| |
|
| |
|
|
| |
|
and 
, prove that
.
