tan (A₁+ A₂+ A₃+…+ A_n) = S₁- S₃ + S₅ – S₅+…/ 1- S₂ + S₄ – S₆+…

where S₁= tan A₁+tan A₂+tan A₃+…+tan A_n,

S₂ = tan A₁ tan A₂+ tan A₂ tan A₃+…And

S₃ = tan A₁ tan A₂ tan A₃ + tan A₂ tan A₃ tan A₄ +…

Dear Sonam

we know

cos(A1+A2+....An)  + i sin(A1+A2 +......An) =(cosA1  + i sinA1)(cosA2  + i sinA2)........(cosAn  + i sinAn)

                                                             =cosA1 cosA2 .....cosAn (1+itanA1)(1+itanA2) .....(1+itanAn)

             now multiply bracket terms

cos(A1+A2+....An)  + i sin(A1+A2 +......An)

=cosA1 cosA2 .....cosAn {1+i(tanA1+tanA2 .......+tanAn) -(tanA1tanA2 + tanA2tanA3.....)-i(S3) +(S4) ........}

compair real and imaginary part

Sin (A₁+ A₂+ A₃+…+ A_n) =cosA1cosA2 ....cosAn( S₁- S₃ + S₅ – S₅+…)

and

Cos(A₁+ A₂+ A₃+…+ A_n) =cosA1cosA2 ....cosAn( 1- S₂ + S₄ – S₆+…)

divide

tan (A₁+ A₂+ A₃+…+ A_n) = S₁- S₃ + S₅ – S₅+…/ 1- S₂ + S₄ – S₆+…

Please feel free to post as many doubts on our discussion forum as you can.
If you find any question Difficult to understand - post it here and we will get you
the answer and detailed  solution very  quickly.

 We are all IITians and here to help you in your IIT JEE preparation.

All the best.
 
Regards,
Askiitians Experts
Badiuddin