To show that [tex]\( 1 - \frac{\cos (2\phi)}{\cos^2 (\phi)} = \tan^2 (\phi) \)[/tex], we will simplify the left-hand side and show that it is equivalent to the right-hand side.
Here's the step-by-step solution:
1. Recall and use trigonometric identities:
- The double-angle identity for cosine: [tex]\(\cos(2\phi) = 2\cos^2(\phi) - 1\)[/tex].
- The definition of tangent: [tex]\(\tan(\phi) = \frac{\sin(\phi)}{\cos(\phi)}\)[/tex].
- The Pythagorean identity: [tex]\(\sin^2(\phi) + \cos^2(\phi) = 1\)[/tex].
2. Substitute the double-angle identity for [tex]\(\cos(2\phi)\)[/tex] in the expression:
[tex]\[
1 - \frac{\cos (2\phi)}{\cos^2 (\phi)}
\][/tex]
becomes
[tex]\[
1 - \frac{2\cos^2(\phi) - 1}{\cos^2(\phi)}.
\][/tex]
3. Simplify the fraction by splitting it into two separate terms:
[tex]\[
1 - \left(\frac{2\cos^2(\phi)}{\cos^2(\phi)} - \frac{1}{\cos^2(\phi)}\right).
\][/tex]
4. Simplify each term inside the parentheses:
- [tex]\(\frac{2\cos^2(\phi)}{\cos^2(\phi)} = 2\)[/tex]
- [tex]\(\frac{1}{\cos^2(\phi)} = \sec^2(\phi)\)[/tex] (since [tex]\(\sec(\phi) = \frac{1}{\cos(\phi)}\)[/tex])
Therefore, the expression becomes:
[tex]\[
1 - (2 - \sec^2(\phi)).
\][/tex]
5. Further simplify the expression:
[tex]\[
1 - 2 + \sec^2(\phi) = \sec^2(\phi) - 1.
\][/tex]
6. Use the Pythagorean identity involving [tex]\(\sec^2(\phi)\)[/tex]:
[tex]\[
\sec^2(\phi) = 1 + \tan^2(\phi).
\][/tex]
7. Substitute the identity into the expression:
[tex]\[
\sec^2(\phi) - 1 = (1 + \tan^2(\phi)) - 1.
\][/tex]
8. Simplify the expression:
[tex]\[
\sec^2(\phi) - 1 = \tan^2(\phi).
\][/tex]
Therefore, we have shown that:
[tex]\[
1 - \frac{\cos (2\phi)}{\cos^2 (\phi)} = \tan^2 (\phi).
\][/tex]
This completes the proof.
