To solve the equation [tex]\(\frac{1}{\cos^2 x} - \tan^2 x = 1\)[/tex], follow these steps:
1. Rewrite the equation using trigonometric identities:
The given equation is [tex]\(\frac{1}{\cos^2 x} - \tan^2 x = 1\)[/tex].
2. Express [tex]\(\tan^2 x\)[/tex] in terms of [tex]\(\sin x\)[/tex] and [tex]\(\cos x\)[/tex]:
Recall that [tex]\(\tan x = \frac{\sin x}{\cos x}\)[/tex]. Therefore, [tex]\(\tan^2 x = \left(\frac{\sin x}{\cos x}\right)^2 = \frac{\sin^2 x}{\cos^2 x}\)[/tex].
3. Substitute [tex]\(\tan^2 x\)[/tex] into the equation:
[tex]\[
\frac{1}{\cos^2 x} - \frac{\sin^2 x}{\cos^2 x} = 1
\][/tex]
4. Combine the terms over a common denominator:
[tex]\[
\frac{1 - \sin^2 x}{\cos^2 x} = 1
\][/tex]
5. Simplify the numerator using a Pythagorean identity:
Recall the identity [tex]\(\sin^2 x + \cos^2 x = 1\)[/tex]. Thus, [tex]\(1 - \sin^2 x = \cos^2 x\)[/tex].
6. Substitute this identity into the equation:
[tex]\[
\frac{\cos^2 x}{\cos^2 x} = 1
\][/tex]
7. Simplify the left-hand side:
[tex]\[
1 = 1
\][/tex]
This verification shows that both sides of the equation are indeed equal after simplification.
So, the given equation simplifies to:
[tex]\[
\frac{1}{\cos^2 x} - \frac{\sin^2 x}{\cos^2 x} = 1
\][/tex]
which simplifies further to:
[tex]\[
1 = 1
\][/tex]
Therefore, the given equation holds true for all [tex]\( x \)[/tex] where [tex]\(\cos x \neq 0\)[/tex] (as [tex]\(\cos^2 x\)[/tex] would be in the denominator and division by zero is undefined).
In conclusion, the equation [tex]\(\frac{1}{\cos^2 x} - \tan^2 x = 1\)[/tex] is an identity that holds for all [tex]\( x \)[/tex] such that [tex]\(\cos x \neq 0\)[/tex].
