Let's start with the given trigonometric equation and simplify it systematically:
[tex]\[ (\sec^2 A - 1) \cos^2 A = \sin^2 A \][/tex]
We need to use trigonometric identities to simplify and verify this equation.
### Step-by-Step Solution:
1. Express Secant in Terms of Cosine:
The secant function is defined as the reciprocal of the cosine function:
[tex]\[ \sec A = \frac{1}{\cos A} \][/tex]
Therefore,
[tex]\[ \sec^2 A = \frac{1}{\cos^2 A} \][/tex]
2. Substitute [tex]\(\sec^2 A\)[/tex] in the Equation:
Replace [tex]\(\sec^2 A\)[/tex] with [tex]\(\frac{1}{\cos^2 A}\)[/tex] in the original equation:
[tex]\[ \left(\frac{1}{\cos^2 A} - 1\right) \cos^2 A = \sin^2 A \][/tex]
3. Simplify Inside the Parentheses:
Now, simplify the expression inside the parentheses:
[tex]\[ \frac{1}{\cos^2 A} - 1 = \frac{1 - \cos^2 A}{\cos^2 A} \][/tex]
4. Multiply by [tex]\(\cos^2 A\)[/tex]:
Multiply each term by [tex]\(\cos^2 A\)[/tex]:
[tex]\[ \left( \frac{1 - \cos^2 A}{\cos^2 A} \right) \cos^2 A = \sin^2 A \][/tex]
By performing the multiplication, we see that the [tex]\(\cos^2 A\)[/tex] terms cancel each other out:
[tex]\[ (1 - \cos^2 A) = \sin^2 A \][/tex]
5. Use the Pythagorean Identity:
Recall the Pythagorean identity for sine and cosine:
[tex]\[ \sin^2 A + \cos^2 A = 1 \][/tex]
Therefore,
[tex]\[ \sin^2 A = 1 - \cos^2 A \][/tex]
6. Verify That Both Sides Match:
Substitute [tex]\(1 - \cos^2 A\)[/tex] on the left side of the simplified equation:
[tex]\[ \sin^2 A = \sin^2 A \][/tex]
This shows that both sides of the equation are identical, which confirms the equation is valid:
[tex]\[ (\sec^2 A - 1) \cos^2 A = \sin^2 A \][/tex]
Thus, the equation holds true and simplifies correctly to zero discrepancy.
