Question

Докажите тождество :
1/(1+ctga) - 1/(1-ctga) = tg 2a

Answer 1

1/(1+ctgα)-1/(1-ctgα)=tg2α
1.  1+ctgα=1+cosα/sinα=(sinα+cosα)/sinα
1:((sinα+cosα)/sinα)=sinα/(sinα+cosα)

2. 1/(1-ctgα)=sinα/(sinα-cosα)

3.   sinα/(sinα+cosα)-sinα/(sinα-cosα)=(sin²α-sinα*cosα-sinα²α-sinα*cosα)/((sinα+cosα)*(sinα-cosα))=(-2sinα*cosα)/(-(cos²α-sin²α))=sin2α/cos2α=tg2α

tg2α=tg2α