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
1056 views around the world
You can reuse this answer
Creative Commons License