1 + tan^2x =
sec^2x
sec^2x
To prove that 1 + tan^2x = sec^2x, we can use trigonometric identities.
First, let us write down the definitions of tangent, secant and cosecant:
tan x = sin x / cos x
sec x = 1 / cos x
csc x = 1 / sin x
Using the above definitions, let’s manipulate the left side of the equation:
1 + tan^2x = 1 + (sin^2x / cos^2x)
= (cos^2x / cos^2x) + (sin^2x / cos^2x)
= (cos^2x + sin^2x) / cos^2x
= 1 / cos^2x
= sec^2x
Therefore, 1 + tan^2x = sec^2x, which is the desired result.
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