Let's solve the equation [tex]\(\sin x \cdot \tan x + \cos x = \frac{1}{\cos x}\)[/tex] step-by-step.
1. Rewrite using trigonometric identities:
Recall that [tex]\(\tan x = \frac{\sin x}{\cos x}\)[/tex], so we can rewrite the given equation:
[tex]\[
\sin x \cdot \frac{\sin x}{\cos x} + \cos x = \frac{1}{\cos x}
\][/tex]
This simplifies to:
[tex]\[
\frac{\sin^2 x}{\cos x} + \cos x = \frac{1}{\cos x}
\][/tex]
2. Combine terms under a common denominator:
The left-hand side has a common denominator of [tex]\(\cos x\)[/tex]. Combine the terms:
[tex]\[
\frac{\sin^2 x + \cos^2 x}{\cos x} = \frac{1}{\cos x}
\][/tex]
3. Simplify using the Pythagorean identity:
Recall that [tex]\(\sin^2 x + \cos^2 x = 1\)[/tex], so:
[tex]\[
\frac{1}{\cos x} = \frac{1}{\cos x}
\][/tex]
This equation holds true for every [tex]\(x\)[/tex] where cos [tex]\(x \neq 0\)[/tex]. However, we also need to ensure the domain and constraints are satisfied.
4. Verify solutions:
Since [tex]\(\cos x = 0\)[/tex] would make the equation undefined as both sides would involve division by zero, we are only left with [tex]\(x\)[/tex] values where [tex]\(\cos x \neq 0\)[/tex]. A common solution where all these values satisfy would be when:
[tex]\[
x = 0
\][/tex]
Therefore, the solution to the given trigonometric equation is:
[tex]\[
x = 0
\][/tex]
This concludes that our step-by-step process has verified [tex]\(x = 0\)[/tex] as the solution.
