Understanding the Pythagorean Identity and Simplifying the Equation 1 + tan^2(x)

1+tan^2(x)=

To solve this equation, we can start by applying the Pythagorean identity which states that 1 + tan^2(x) = sec^2(x)

To solve this equation, we can start by applying the Pythagorean identity which states that 1 + tan^2(x) = sec^2(x).

The Pythagorean identity, sin^2(x) + cos^2(x) = 1, can be rewritten in terms of tangents as sin^2(x)/cos^2(x) + 1 = 1/cos^2(x).

Using the quotient identity, sin^2(x)/cos^2(x) = tan^2(x), we can substitute this into the equation to get tan^2(x) + 1 = 1/cos^2(x).

Rearranging the equation, we have tan^2(x) = 1/cos^2(x) – 1.

To simplify further, we can find a common denominator for the two fractions. The denominator of 1 in the second fraction is cos^2(x).

So the equation becomes tan^2(x) = (1 – cos^2(x))/cos^2(x).

Using the identity sin^2(x) = 1 – cos^2(x), we can rewrite the numerator as sin^2(x).

Thus, the equation simplifies to tan^2(x) = sin^2(x)/cos^2(x).

And finally, using the definition of the tangent function, we know that tan(x) = sin(x)/cos(x).

So tan^2(x) simplifies to (sin(x)/cos(x))^2.

Therefore, the solution to the equation 1 + tan^2(x) is simply tan^2(x).

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