Sure! Let's solve the equation step-by-step:
[tex]\[
\sec^2(x) \left(1 - \sin^2(x)\right) = 1
\][/tex]
### Step 1: Apply the Pythagorean Identity
First, recall the Pythagorean identity:
[tex]\[
\sin^2(x) + \cos^2(x) = 1
\][/tex]
From this identity, we can express [tex]\( \cos^2(x) \)[/tex] as:
[tex]\[
\cos^2(x) = 1 - \sin^2(x)
\][/tex]
So, substitute [tex]\( \cos^2(x) \)[/tex] for [tex]\( 1 - \sin^2(x) \)[/tex]:
[tex]\[
\sec^2(x) \cdot \cos^2(x) = 1
\][/tex]
### Step 2: Use the Definition of Secant
Recall the definition of [tex]\( \sec(x) \)[/tex]:
[tex]\[
\sec(x) = \frac{1}{\cos(x)}
\][/tex]
Therefore,
[tex]\[
\sec^2(x) = \left(\frac{1}{\cos(x)}\right)^2 = \frac{1}{\cos^2(x)}
\][/tex]
### Step 3: Simplify the Equation
Substitute [tex]\( \frac{1}{\cos^2(x)} \)[/tex] for [tex]\( \sec^2(x) \)[/tex]:
[tex]\[
\frac{1}{\cos^2(x)} \cdot \cos^2(x) = 1
\][/tex]
Now, multiply:
[tex]\[
\left( \frac{1}{\cos^2(x)} \right) \cos^2(x) = 1
\][/tex]
### Step 4: Simplify Further
The [tex]\( \cos^2(x) \)[/tex] terms cancel each other out:
[tex]\[
1 = 1
\][/tex]
### Conclusion
The equation is an identity, meaning it holds true for all [tex]\( x \)[/tex] where it is defined. Specifically, it holds true for all real numbers [tex]\( x \)[/tex] except for those values where [tex]\( \cos(x) = 0 \)[/tex] (since [tex]\( \sec(x) \)[/tex] would be undefined).
Thus, the given equation [tex]\( \sec^2(x) \left(1 - \sin^2(x)\right) = 1 \)[/tex] is an identity and holds true for all real numbers [tex]\( x \)[/tex] where [tex]\( x \)[/tex] is defined.
