Discover The Trigonometric Identity: Rearranging Tan^2X + 1 = Sec^2X To Get 1 + Tan^2X

1+tan^2x

sec^2x

We know that tan^2x + 1 = sec^2x.

Rearrange this equation to obtain 1 + tan^2x on one side:

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

Now substitute 1/cos^2x into the equation:

tan^2x = (1/cos^2x) – 1
tan^2x = (1 – cos^2x) / cos^2x

We know that 1 – cos^2x = sin^2x, so we can substitute that in:

tan^2x = (1 – cos^2x) / cos^2x
tan^2x = sin^2x / cos^2x

Finally, remember that tanx = sinx/cosx, so substitute that in:

tan^2x = sin^2x / cos^2x
tan^2x = (tanx)^2

Therefore,

1 + tan^2x = 1 + (tanx)^2

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