Certainly! Let's go through the steps to prove that [tex]\(\cos 2\theta = \sin 4\phi\)[/tex] given [tex]\(\tan \theta = \frac{1}{7}\)[/tex] and [tex]\(\tan \phi = \frac{1}{3}\)[/tex].
### Step-by-Step Solution:
1. Represent [tex]\(\theta\)[/tex] and [tex]\(\phi\)[/tex] using arctangent:
Since [tex]\(\tan \theta = \frac{1}{7}\)[/tex], we have:
[tex]\[
\theta = \tan^{-1}\left(\frac{1}{7}\right)
\][/tex]
Similarly, since [tex]\(\tan \phi = \frac{1}{3}\)[/tex], we have:
[tex]\[
\phi = \tan^{-1}\left(\frac{1}{3}\right)
\][/tex]
2. Double-angle formula for cosine:
For [tex]\(\cos 2\theta\)[/tex], we use the double-angle identity for cosine in terms of tangent:
[tex]\[
\cos 2\theta = \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta}
\][/tex]
Substituting [tex]\(\tan \theta = \frac{1}{7}\)[/tex]:
[tex]\[
\cos 2\theta = \frac{1 - \left(\frac{1}{7}\right)^2}{1 + \left(\frac{1}{7}\right)^2} = \frac{1 - \frac{1}{49}}{1 + \frac{1}{49}} = \frac{\frac{48}{49}}{\frac{50}{49}} = \frac{48}{50} = \frac{24}{25}
\][/tex]
Thus,
[tex]\[
\cos 2\theta = 0.96
\][/tex]
3. Double-angle formula for sine:
To find [tex]\(\sin 4\phi\)[/tex], use the identity:
[tex]\[
\sin 4\phi = 2 \sin 2\phi \cos 2\phi
\][/tex]
First, find [tex]\(\sin 2\phi\)[/tex] and [tex]\(\cos 2\phi\)[/tex] using the double-angle formulas for sine and cosine:
[tex]\[
\sin 2\phi = \frac{2 \tan \phi}{1 + \tan^2 \phi}
\][/tex]
Substituting [tex]\(\tan \phi = \frac{1}{3}\)[/tex]:
[tex]\[
\sin 2\phi = \frac{2 \left(\frac{1}{3}\right)}{1 + \left(\frac{1}{3}\right)^2} = \frac{\frac{2}{3}}{1 + \frac{1}{9}} = \frac{\frac{2}{3}}{\frac{10}{9}} = \frac{2}{3} \times \frac{9}{10} = \frac{6}{10} = \frac{3}{5}
\][/tex]
[tex]\[
\cos 2\phi = \frac{1 - \tan^2 \phi}{1 + \tan^2 \phi} = \frac{1 - \left(\frac{1}{3}\right)^2}{1 + \left(\frac{1}{3}\right)^2} = \frac{1 - \frac{1}{9}}{1 + \frac{1}{9}} = \frac{\frac{8}{9}}{\frac{10}{9}} = \frac{8}{10} = \frac{4}{5}
\][/tex]
Now, calculate [tex]\(\sin 4\phi\)[/tex]:
[tex]\[
\sin 4\phi = 2 \sin 2\phi \cos 2\phi = 2 \left(\frac{3}{5}\right) \left(\frac{4}{5}\right) = 2 \times \frac{12}{25} = \frac{24}{25}
\][/tex]
Thus,
[tex]\[
\sin 4\phi = 0.96
\][/tex]
4. Conclusion:
We have found that:
[tex]\[
\cos 2\theta = 0.96 \quad \text{and} \quad \sin 4\phi = 0.96
\][/tex]
Therefore,
[tex]\[
\cos 2\theta = \sin 4\phi
\][/tex]
Hence, it is proven that [tex]\(\cos 2\theta = \sin 4\phi\)[/tex].
