Saturday, 4 June 2016

trigonometry - Proving tan(theta1+theta2+cdots+thetan) has the form fracS1S3+S5+cdots1S2+S4+cdots



tan(θ1+θ2++θn)=S1S3+S5+1S2+S4+



where Si denotes the sum of product of tangent of angles taken i at a time.





For example,tan(θ1+θ2)=S11S2=tanθ1+tanθ21tanθ1tanθ2




(This formula is given in my textbook with no derivation or background)



How to derive this?


Answer




Prove that is true for 2 angles, then consider it true for (n1) angles and prove for n angles.
If you prove it for 2 angles, it will be easy to prove it for n angles (knowing it is true for (n1) angles) by considering θ1+...+θn1 is one angle, and θn is the other.


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