If `cos^4(A) – sin^4(A) = p`, then find the value of `p`.
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Solution
Given:
\[ \cos^4 A – \sin^4 A = p \]
Step 1: Use the difference of squares formula
\[ \cos^4 A – \sin^4 A = (\cos^2 A + \sin^2 A)(\cos^2 A – \sin^2 A) \]
Step 2: Use the Pythagorean identity
Since \(\sin^2 A + \cos^2 A = 1\), we have:
\[ (\cos^2 A + \sin^2 A)(\cos^2 A – \sin^2 A) = (1)(\cos^2 A – \sin^2 A) \]
Step 3: Use the double angle formula
The double angle formula for cosine is \(\cos 2A = \cos^2 A – \sin^2 A\), so:
\[ \cos^2 A – \sin^2 A = \cos 2A \]
Step 4: Find the value of \(p\)
Therefore, we have:
\[ p = \cos 2A \]
Conclusion
The value of \(p\) is \(\cos 2A\).