Simplifying the Expression 1 + tan^2(x) using the Pythagorean Identity

1 + tan^2 x

To simplify the expression 1 + tan^2(x), we can use the Pythagorean identity for trigonometric functions

To simplify the expression 1 + tan^2(x), we can use the Pythagorean identity for trigonometric functions. The Pythagorean identity states that for any angle x, sin^2(x) + cos^2(x) = 1.

We can rewrite tan^2(x) as (sin^2(x))/(cos^2(x)). Using the Pythagorean identity, we know that sin^2(x) + cos^2(x) = 1. Therefore, we have:

tan^2(x) = (sin^2(x))/(cos^2(x))

Now, let’s substitute this into the original expression:

1 + tan^2(x) = 1 + (sin^2(x))/(cos^2(x))

To combine these terms, we need a common denominator. In this case, the common denominator is cos^2(x). Therefore, we multiply 1 by cos^2(x)/cos^2(x) to get:

1 + tan^2(x) = (cos^2(x))/(cos^2(x)) + (sin^2(x))/(cos^2(x))

Now, we can combine the terms over the common denominator:

1 + tan^2(x) = (cos^2(x) + sin^2(x))/(cos^2(x))

Using the Pythagorean identity again, we know that cos^2(x) + sin^2(x) = 1. So, we can simplify further:

1 + tan^2(x) = 1/(cos^2(x))

Therefore, the simplified expression for 1 + tan^2(x) is 1/(cos^2(x)).

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