Understanding the Simplified Expression of sec^2(x) in Trigonometry

int sec^2(x)

The expression “sec^2(x)” represents the square of the secant function of angle x

The expression “sec^2(x)” represents the square of the secant function of angle x. To find a more simplified expression, we can use trigonometric identities.

Recall that sec(x) is the reciprocal of cosine(x). So, we can rewrite the expression as follows:

sec^2(x) = (1/cos(x))^2

To square the reciprocal, we multiply the numerator and denominator by itself:

sec^2(x) = (1^2)/(cos^2(x)) = 1/cos^2(x)

Now, we can simplify further by using the identity cos^2(x) + sin^2(x) = 1. Rearranging this equation, we get cos^2(x) = 1 – sin^2(x). Substituting this into our expression:

sec^2(x) = 1/(1 – sin^2(x))

Applying another trigonometric identity, sin^2(x) + cos^2(x) = 1, we can rewrite the expression as:

sec^2(x) = 1/(cos^2(x)) = 1/(1 – sin^2(x)) = 1/(1 – (1 – cos^2(x))) = 1/(1 – 1 + cos^2(x)) = 1/(cos^2(x))

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

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