that): \( \frac{1-\sin ^{4} A}{\cos ^{4} A}=1+2 \tan ^{2} A \)

Certainly! Let's **prove the trigonometric identity**: \[ \frac{1 - \sin^4 A}{\cos^4 A} = 1 + 2 \tan^2 A \] ### **Step 1: Simplify the Numerator** Start by simplifying the numerator \(1 - \sin^4 A\). Notice that \(1 - \sin^4 A\) is a **difference of squares**, which can be factored as: \[ 1 - \sin^4 A = (1 - \sin^2 A)(1 + \sin^2 A) \] ### **Step 2: Apply the Pythagorean Identity** Recall the Pythagorean identity: \[ \sin^2 A + \cos^2 A = 1 \quad \Rightarrow \quad 1 - \sin^2 A = \cos^2 A \] Substitute this into the factored form: \[ 1 - \sin^4 A = \cos^2 A (1 + \sin^2 A) \] ### **Step 3: Rewrite the Original Expression** Now, substitute the simplified numerator back into the original expression: \[ \frac{1 - \sin^4 A}{\cos^4 A} = \frac{\cos^2 A (1 + \sin^2 A)}{\cos^4 A} \] ### **Step 4: Simplify the Fraction** Divide the numerator and the denominator by \(\cos^2 A\): \[ \frac{\cos^2 A (1 + \sin^2 A)}{\cos^4 A} = \frac{1 + \sin^2 A}{\cos^2 A} \] ### **Step 5: Separate the Terms** Break down the fraction into two separate terms: \[ \frac{1}{\cos^2 A} + \frac{\sin^2 A}{\cos^2 A} \] ### **Step 6: Apply Trigonometric Identities** Recall the following trigonometric identities: \[ \frac{1}{\cos^2 A} = \sec^2 A \] \[ \frac{\sin^2 A}{\cos^2 A} = \tan^2 A \] Substitute these into the expression: \[ \sec^2 A + \tan^2 A \] ### **Step 7: Use the Pythagorean Identity Again** Another Pythagorean identity states: \[ \sec^2 A = 1 + \tan^2 A \] Substitute this into the expression: \[ (1 + \tan^2 A) + \tan^2 A = 1 + 2 \tan^2 A \] ### **Conclusion** Putting it all together, we've shown that: \[ \frac{1 - \sin^4 A}{\cos^4 A} = 1 + 2 \tan^2 A \] This completes the proof of the identity.