To simplify the expression [tex]\(\cos x \left(1 + \tan^2 x \right)\)[/tex], we can make use of some trigonometric identities.
First, recall the trigonometric identity for [tex]\(\tan^2 x\)[/tex]:
[tex]\[
\tan^2 x = \frac{\sin^2 x}{\cos^2 x}
\][/tex]
Using this identity, we can rewrite the original expression:
[tex]\[
\cos x \left(1 + \tan^2 x \right) = \cos x \left(1 + \frac{\sin^2 x}{\cos^2 x} \right)
\][/tex]
Next, we combine the terms inside the parentheses by getting a common denominator:
[tex]\[
1 + \frac{\sin^2 x}{\cos^2 x} = \frac{\cos^2 x}{\cos^2 x} + \frac{\sin^2 x}{\cos^2 x} = \frac{\cos^2 x + \sin^2 x}{\cos^2 x}
\][/tex]
We then use the Pythagorean identity:
[tex]\[
\cos^2 x + \sin^2 x = 1
\][/tex]
This simplifies our expression to:
[tex]\[
\frac{1}{\cos^2 x}
\][/tex]
Now, we substitute this back into the original context:
[tex]\[
\cos x \left(\frac{1}{\cos^2 x}\right)
\][/tex]
Simplifying further by canceling [tex]\(\cos x\)[/tex] in the numerator and denominator, we get:
[tex]\[
\frac{1}{\cos x}
\][/tex]
Thus, the simplified form of the expression is:
[tex]\[
\boxed{\frac{1}{\cos x}}
\][/tex]
