Asked by Lucia
How can I prove this identity?
(1 + sinØ + cosØ):(1 - sinØ + cosØ) =
(1 + sinØ) : cosØ
(1 + sinØ + cosØ):(1 - sinØ + cosØ) =
(1 + sinØ) : cosØ
Answers
Answered by
MathMate
Work on the left hand side:
(1 + sinØ + cosØ)/(1 - sinØ + cosØ)
multiply both the numerator and denominator by (1-sinφ+cosφ) and simplify:
((1+cosφ)^2-sin&hi;^2)/(1-sinφ+cosφ)^2
=2cosφ(1+cosφ)/[2(1-sinφ)(1+cosφ)]
=2cosφ/[2(1-sinφ)]
=cosφ/(1-sinφ)
Now multiply top and bottom by (1+sinφ)
cosφ(1+sinφ)/(1-sin²φ)
=cosφ(1+sinφ)/cos²φ
=(1+sinφ)/cosφ
(1 + sinØ + cosØ)/(1 - sinØ + cosØ)
multiply both the numerator and denominator by (1-sinφ+cosφ) and simplify:
((1+cosφ)^2-sin&hi;^2)/(1-sinφ+cosφ)^2
=2cosφ(1+cosφ)/[2(1-sinφ)(1+cosφ)]
=2cosφ/[2(1-sinφ)]
=cosφ/(1-sinφ)
Now multiply top and bottom by (1+sinφ)
cosφ(1+sinφ)/(1-sin²φ)
=cosφ(1+sinφ)/cos²φ
=(1+sinφ)/cosφ
There are no AI answers yet. The ability to request AI answers is coming soon!
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.