sec^2x
1+tan^2x
The expression sec^2x is the square of the secant function of angle x. The secant function is the reciprocal of the cosine function, so we can write:
sec x = 1 / cos x
Therefore, we can square both sides to get:
sec^2 x = (1 / cos x)^2
Expanding the denominator on the right-hand side gives:
sec^2 x = 1 / (cos x)^2
Using the identity cos^2 x + sin^2 x = 1, we can substitute sin^2 x for 1 – cos^2 x to get:
sec^2 x = 1 / (cos x)^2
= 1 / ((1 – sin^2 x)^2)
= 1 / (1 – 2 sin^2 x + sin^4 x)
This is an alternative form for sec^2 x in terms of the sine function. You can simplify it further by using the Pythagorean identity sin^2 x + cos^2 x = 1 to eliminate sin^4 x from the expression.
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