Question

Find the value of tan 100

tan20degree

tan40degree

tan50

tan 70

tan80

. ​

Answer 1

Answer:1

Step-by-step explanation:

Gauss’s first grade teacher, so the story goes, asked the class to add the numbers from 1 to 100 to keep them busy for a while. Gauss quickly wrote 5050 on his slate, realizing the sum is easier when calculated as

(1+100)+(2+99)+(3+98)+…+(50+51)=50(101)  

A similar trick works here. The tangents of complementary angles are reciprocals, as this shows:

tan(90∘−x)=cotx=1/tanx  

So

tan10∘tan20∘tan30∘tan40∘tan50∘tan60∘tan70∘tan80∘  

=(tan10∘tan80∘)(tan20∘tan70∘)(tan30∘tan60∘)(tan40∘tan50∘)  

=(1)(1)(1)(1)=1

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Answer 2

Find the value of tan 100

tan20degree

tan40degree

tan50

tan 70

tan80