Proving an identity and solving an equation?

(i) Prove the identity 1/costheta - costheta/(1 + sintheta) \equiv tantheta

(ii) Solve the equation 1/costheta - costheta/(1 + sintheta) + 2 = 0 for 0 degrees <= theta <= 360 degrees.

2 Answers
Jim G.
Aug 7, 2017

"see explanation"

Explanation:

"consider the left side"

1/costheta-(costheta)/(1+sintheta)larr" express as single fraction"

=(1+sintheta)/(costheta(1+sintheta))-(cos^2theta)/(costheta(1+sintheta))

=(1+sintheta-cos^2theta)/(costheta(1+sintheta))

=(1+sintheta-(1-sin^2theta))/(costheta(1+sintheta))

=(sinthetacancel((1+sintheta)))/(costhetacancel((1+sintheta)))

=sintheta/costheta=tantheta=" right side "rArr" proved"

"for "(ii)" we now have"

tantheta+2=0

rArrtantheta=-2

tantheta" is negative in second and fourth quadrants"

rArrtheta=tan^-1(2)=63.43^@larrcolor(red)" related acute angle"

rArrtheta=(180-63.43)^@=116.57^@

"or "theta=(360-63.43)^@=296.57^@

rArrtheta=116.57^@,296.57^@to0<=theta<=360

P dilip_k
Aug 7, 2017

LHS=1/costheta-costheta/(1+sintheta)

=1/costheta-((1-sintheta)costheta)/((1-sintheta)(1+sintheta))

=sectheta-((1-sintheta)costheta)/(1-sin^2theta)

=sectheta-((1-sintheta)costheta)/cos^2theta

=sectheta-(1-sintheta)/costheta

=sectheta-1/costheta+sintheta/costheta

=sectheta-sectheta+sintheta/costheta

=tantheta=RHS

Related questions
Impact of this question
1173 views around the world
You can reuse this answer
Creative Commons License