# Simplify and express as a constant or a single function of the given angle. Show all work (a) 1-sin^2 Pheta/ Cos^2 Pheta (b) Csc3PhetaTan3PhetaCos3Pheta

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A) we have given

1-sin^2(ptheta)/cos^2(ptheta)

(cos^2(ptheta)-sin^2(ptheta))/cos^2(ptheta)

=cos(2ptheta)/cos^2(ptheta)

=Cos(2p theta)sec^2(ptheta)

B). Cosec(3ptheta)tan(3ptheta)cos(3ptheta)

(1/sin(3ptheta))(sin(3ptheta)/cos(3ptheta))cos(3ptheta)

=1

Simplify:

(1) `(1-sin^2theta)/(cos^2 theta)`

From the Pythagorean identity `sin^2theta+cos^2theta=1` we get `cos^2theta=1-sin^2theta`

Then `(1-sin^2theta)/(cos^2theta)=(cos^2theta)/(cos^2theta)=1`

(2) `csc3thetatan3thetacos3theta`

We use the reciprocal identities: `cscx=1/(sinx),tanx=(sinx)/(cosx)` . Substituting we get:

`1/(sin3theta)*(sin3theta)/(cos3theta)*cos3theta=1`