Вычислите: tg 1⁰ * tg 2⁰ * tg 3⁰ * ..... * tg 89⁰

Question

Вычислите:

tg 1⁰ * tg 2⁰ * tg 3⁰ * ..... * tg 89⁰

Answer 1

tg1*tg2*....tg88*tg89=(tg1*tg89)*(tg2*tg88)....*(tg44*tg46)*tg45=

=(cos(88-cos90)/(cos88+cos90))*(cos(86-cos90)/(cos86+cos90))*...((cos(2-cos90)/(cos2+cos90))*tg45=

=cos88/cos88 + cos86/cos86*....*cos2/cos2 *tg45=1

Answer 2

 tg 1⁰ * tg 2⁰ * tg 3⁰ * ..... * tg 89⁰=

= tg 1⁰ * tg 2⁰ * tg 3⁰ * ...tg 44⁰*tg45⁰*tg46⁰.. * tg 89⁰=

=tg 1⁰ * tg 2⁰ * tg 3⁰ * ...tg 44⁰*tg45⁰*ctg44⁰.. * ctg 1⁰=

=(tg 1⁰* ctg 1⁰)*(tg 2⁰* ctg 2⁰)*...*(tg 44⁰* ctg 44⁰)*tg45⁰=

=1*1*..*1*1=1

ответ: 1

 

так как tg(90⁰-x)=ctg x

tg x * ctg x=1