1 + tan^2x =
sec^2x
sec^2x
To prove that 1 + tan^2x = sec^2x, we will use the trigonometric identity:
sec^2x = 1 + tan^2x
Now, let’s simplify 1 + tan^2x to see if it matches the right-hand side of the equation.
1 + tan^2x = (1/cos^2x) + (sin^2x/cos^2x) (By dividing sine with cosine.)
= [(1 + sin^2x)/cos^2x]
We know that 1 + sin^2x = cos^2x (by using Pythagorean identity sin^2x + cos^2x = 1)
Therefore, 1 + tan^2x = (1/cos^2x) + (sin^2x/cos^2x) = (cos^2x/cos^2x) = sec^2x
Hence, we have proved that 1 + tan^2x = sec^2x.
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