Let’s verify the given trigonometric identity step by step:
[tex]\[ \frac{1 - \sin^4 A}{\cos^4 A} = 1 + 2 \tan^2 A \][/tex]
### Step 1: Simplify the Left-Hand Side
We start with:
[tex]\[ \frac{1 - \sin^4 A}{\cos^4 A} \][/tex]
Using the identity [tex]\( \sin^2 A + \cos^2 A = 1 \)[/tex], we know that [tex]\( \sin^2 A = 1 - \cos^2 A \)[/tex]. Consequently, we can express [tex]\( \sin^4 A \)[/tex] as:
[tex]\[ \sin^4 A = (\sin^2 A)^2 = (1 - \cos^2 A)^2 \][/tex]
Therefore, the expression becomes:
[tex]\[ \frac{1 - \sin^4 A}{\cos^4 A} = \frac{1 - (1 - \cos^2 A)^2}{\cos^4 A} \][/tex]
### Step 2: Expand and Simplify the Numerator
Expand [tex]\( (1 - \cos^2 A)^2 \)[/tex]:
[tex]\[ (1 - \cos^2 A)^2 = 1 - 2\cos^2 A + \cos^4 A \][/tex]
Substitute back into the numerator:
[tex]\[ 1 - (1 - 2\cos^2 A + \cos^4 A) = 1 - 1 + 2\cos^2 A - \cos^4 A \][/tex]
[tex]\[ = 2\cos^2 A - \cos^4 A \][/tex]
Therefore, the expression now is:
[tex]\[ \frac{2\cos^2 A - \cos^4 A}{\cos^4 A} \][/tex]
### Step 3: Simplify the Fraction
Separate the terms in the numerator:
[tex]\[ \frac{2\cos^2 A - \cos^4 A}{\cos^4 A} = \frac{2\cos^2 A}{\cos^4 A} - \frac{\cos^4 A}{\cos^4 A} \][/tex]
[tex]\[ = \frac{2}{\cos^2 A} - 1 \][/tex]
Recognizing that:
[tex]\[ \frac{1}{\cos^2 A} = \sec^2 A \][/tex]
[tex]\[ \frac{2}{\cos^2 A} = 2\sec^2 A \][/tex]
So the left-hand side simplifies to:
[tex]\[ 2\sec^2 A - 1 \][/tex]
### Step 4: Simplify the Right-Hand Side
We note that:
[tex]\[ \tan^2 A = \frac{\sin^2 A}{\cos^2 A} \][/tex]
So,
[tex]\[ 1 + 2\tan^2 A = 1 + 2 \frac{\sin^2 A}{\cos^2 A} \][/tex]
Using the Pythagorean identity, we know:
[tex]\[ \sec^2 A = 1 + \tan^2 A \][/tex]
Thus,
[tex]\[ \sec^2 A = \frac{1}{\cos^2 A} \][/tex]
Hence,
[tex]\[ 1 + 2\tan^2 A = 1 + 2(\frac{\sin^2 A}{\cos^2 A}) = 2\sec^2 A - 1 \][/tex]
### Conclusion
Both the left-hand side and right-hand side simplify to:
[tex]\[ 2\sec^2 A - 1 \][/tex]
Therefore, the given identity holds true:
[tex]\[ \frac{1 - \sin^4 A}{\cos^4 A} = 1 + 2\tan^2 A \][/tex]
