Let's solve the given trigonometric equation step-by-step:
[tex]\[ 4\left(\frac{\cos^4 A + \sin^2 A}{3 + \cos 4A}\right) = (\cos^4 A - \sin^4 A) \sec 2A \][/tex]
Step 1: Start by simplifying the left-hand side (LHS) of the equation:
[tex]\[ \text{LHS} = 4\left(\frac{\cos^4 A + \sin^2 A}{3 + \cos 4A}\right) \][/tex]
Step 2: Now we simplify the right-hand side (RHS) of the equation:
[tex]\[ \text{RHS} = (\cos^4 A - \sin^4 A) \sec 2A \][/tex]
[tex]\[ \text{Since } \sec 2A = \frac{1}{\cos 2A} \][/tex]
[tex]\[ \text{RHS} = \frac{\cos^4 A - \sin^4 A}{\cos 2A} \][/tex]
Step 3: Next, evaluate specific trigonometric identities and values to help analyze the expressions.
To compare LHS and RHS more easily, we note specific properties and identities, specifically examining [tex]\(\cos 4A\)[/tex] in LHS.
Step 4: Determine if the simplified LHS and RHS are equivalent.
After mathematical manipulations and simplifications:
[tex]\[ LHS = 4\left(\frac{\cos^4 A + \sin^2 A}{3 + \cos 4A}\right) \][/tex]
[tex]\[ RHS = \frac{\cos^4 A - \sin^4 A}{\cos 2A} \][/tex]
Step 5: Compare the simplified forms of LHS and RHS directly.
It turns out that:
[tex]\[ 4\left(\frac{\sin^2 A + \cos^4 A}{3 + \cos 4A}\right) \][/tex]
is not identical to:
[tex]\[ 1 \][/tex]
Thus, the left-hand side does not equal the right-hand side. As a result, the expression [tex]\(4\left(\frac{\cos ^4 A+\sin ^2 A}{3+\cos 4 A}\right) = (\cos ^4 A-\sin ^4 A) \sec 2 A\)[/tex] does not hold true.
