To prove the trigonometric identity
[tex]\[
\frac{1}{\cos x} - \sin x \tan x = \cos x,
\][/tex]
we will simplify the left-hand side (LHS) of the equation step by step until it matches the right-hand side (RHS).
Given:
[tex]\[
\frac{1}{\cos x} - \sin x \tan x = \cos x.
\][/tex]
Step 1: Start with the left-hand side (LHS):
[tex]\[
\frac{1}{\cos x} - \sin x \tan x.
\][/tex]
Step 2: Recall that [tex]\(\tan x = \frac{\sin x}{\cos x}\)[/tex], and substitute this expression into the LHS:
[tex]\[
\frac{1}{\cos x} - \sin x \left(\frac{\sin x}{\cos x}\right).
\][/tex]
Step 3: Simplify the expression inside the parentheses:
[tex]\[
\frac{1}{\cos x} - \frac{\sin^2 x}{\cos x}.
\][/tex]
Step 4: Combine the fractions over a common denominator:
[tex]\[
\frac{1 - \sin^2 x}{\cos x}.
\][/tex]
Step 5: Use the Pythagorean identity, which states that [tex]\( \sin^2 x + \cos^2 x = 1 \)[/tex]. This implies that [tex]\( 1 - \sin^2 x = \cos^2 x \)[/tex]:
[tex]\[
\frac{\cos^2 x}{\cos x}.
\][/tex]
Step 6: Simplify the fraction:
[tex]\[
\cos x.
\][/tex]
Conclusion:
We have simplified the left-hand side (LHS) to match the right-hand side (RHS):
[tex]\[
\cos x = \cos x.
\][/tex]
Thus, the identity is verified:
[tex]\[
\frac{1}{\cos x} - \sin x \tan x = \cos x.
\][/tex]
