To verify the trigonometric identity [tex]\(\frac{\sin^2 x + \cos^2 x}{\cos x} = \sec x\)[/tex], we can use the Pythagorean identity. The Pythagorean identity in trigonometry states:
[tex]\[
\sin^2 x + \cos^2 x = 1
\][/tex]
Here's a step-by-step verification of the given identity using this basic trigonometric identity:
1. Start with the left-hand side of the given identity:
[tex]\[
\frac{\sin^2 x + \cos^2 x}{\cos x}
\][/tex]
2. Substitute the Pythagorean identity [tex]\(\sin^2 x + \cos^2 x = 1\)[/tex] into the expression:
[tex]\[
\frac{1}{\cos x}
\][/tex]
3. Recognize that [tex]\(\frac{1}{\cos x}\)[/tex] is the definition of [tex]\(\sec x\)[/tex] (secant of x):
[tex]\[
\sec x
\][/tex]
4. Therefore, the left-hand side simplifies to:
[tex]\[
\sec x
\][/tex]
This shows that the left-hand side is equal to the right-hand side. Hence, the identity [tex]\(\frac{\sin^2 x + \cos^2 x}{\cos x} = \sec x\)[/tex] is verified.
Thus, [tex]\(\cos^2 x + \sin^2 x = 1\)[/tex] is the basic trigonometric identity used to verify the given equation.
