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# Trig identity question.?

1+tan^2x=1/cos^2x

how?

### 3 Answers

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• Favorite Answer

LHS = 1 + tan^2 x

LHS = (cos^2 x + sin^2 x) / cos^2 x

LHS = 1 / cos^2 x

LHS = RHS

QED

• Take the Pythagorean Identity and divide both sides by cos^2 x.

Pythagorean Identity: (sin^2 x +cos^2 x = 1).

• I am assuming you want to prove this identity. First, express everything in terms of sine and cosine.

LS = 1 + tan^2(x)

LS = 1 + sin^2(x) / cos^2(x)

RS = 1 / cos^2(x)

Next I when I start seeing sin^2 and cos^2 in a proof, I start thinking of using the Pythagorean identity. sin^2(x) + cos^2(x) = 1, for all x.

To perform the addition on the left side, I need a common denominator, which in this case is cos^2(x).

LS = cos^2(x)/cos^2(x) + sin^2(x) / cos^2(x)

LS = [cos^2(x) + sin^2(x) ] / cos^2(x)

Apply the Pythagorean identity to the numerator.

LS = 1 / cos^2(x)

LS = RS

Therefore, proven.

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