How does sec-squared x equals 1 plus tan-squared x?

Answer 1

Let s = sin x; c = cos x.

By definition,

sec x = 1/cos x = 1/c; and

tan x = (sin x) / (cos x) = s/c.

We know, also, that s2 + c2 = 1.

Then, dividing through by c2, we have,

(s2/c2) + 1 = (1/c2), or

(s/c)2 + 1 = (1/c)2; in other words, we have,

tan2 x + 1 = sec2 x.

Answer 2

Original Equation

sec2(x) = 1 + tan2(x)Leave the left side assec2(x) and for the proof, try to make the right side equal the left side.

We know tan = sin/cos

1 + tan2(x) = 1 + sin2(x)/cos2(x)

Rewrite the 1 in terms of cos2(x)

1 + sin2(x)/cos2(x) = cos2(x)/cos2(x) + sin2(x)/cos2(x)

Simplify, now that the denominators have the same terms

cos2(x)/cos2(x) + sin2(x)/cos2(x) = [ cos2(x) + sin2(x) ] / cos2(x)

Use the trig. identity: cos2(x) + sin2(x) = 1

[ cos2(x) + sin2(x) ] / cos2(x) = 1 / cos2(x)

Remember 1/cosine is equal to secant.\

1 / cos2(x) = sec2(x)

Recall, the left side the equation was: sec2(x)
The right side (we just solved for is): sec2(x)

sec2(x) = sec2(x)
Left side = right side, Q.E.D.