Question #17801

How do i prove tan(theta)+cot(theta)/sec(theta)csc(theta)=1?

Expert's answer

sec(theta)=1/cos(theta)

csc(theta)=1/sin(theta)

tan(theta)=sin(theta)/cos(theta)

cot(theta)=cos(theta)/sin(theta)

( tan(theta)+cot(theta) )/( sec(theta)csc(theta))=[sin(theta)/cos(theta)+cos(theta)/sin(theta)]/[1/( sin(theta) * cos(theta) ) ]=

=[ (sin^2(theta)+cos^2 (theta) )/( sin(theta)cos(theta) )

]*[sin(theta)*cos(theta) ]=sin^2(theta)+cos^2 (theta)=1

csc(theta)=1/sin(theta)

tan(theta)=sin(theta)/cos(theta)

cot(theta)=cos(theta)/sin(theta)

( tan(theta)+cot(theta) )/( sec(theta)csc(theta))=[sin(theta)/cos(theta)+cos(theta)/sin(theta)]/[1/( sin(theta) * cos(theta) ) ]=

=[ (sin^2(theta)+cos^2 (theta) )/( sin(theta)cos(theta) )

]*[sin(theta)*cos(theta) ]=sin^2(theta)+cos^2 (theta)=1

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