Question #dcdc4

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Answer 1 · 2017-08-01T19:59:08

Start with:

$cos^2(theta) + sin^2(theta) = 1$

Divide both sides of the equation by $cos^2(theta)$

$1 + sin^2(theta)/cos^2(theta) = 1/cos^2(theta)$

Substitute $tan^2(theta)$ for $sin^2(theta)/cos^2(theta)$ :

$1 + tan^2(theta) = 1/cos^2(theta)$

Substitute $sec^2(theta)$ for $1/cos^2(theta)$ :

$1 + tan^2(theta) = sec^2(theta)$ Q.E.D.

Answer 2 · 2017-08-01T20:02:09

See the proof below

We need

$tantheta=sintheta/costheta$

$sectheta=1/costheta$

We start with

$sin^2theta+cos^2theta=1$

Dividing throughout by $cos^2theta$

$sin^2theta/cos^2theta+cos^2theta/cos^2theta=1/cos^2theta$

$tan^2theta+1=sec^2theta$

$QED$