Question

tan3 + tan6 + tan9 +........+tan89 = 1 prove it pls

Answer 1

  No, It is not equal to 1,.
We know that,
value of  tan θ when, θ < 90° is always positive,.
hence, we know that,

⇒ Sum of all positive integers is always positive,.
&
Tan 60° = √3,.
√3 > 1,.
Hence,.
Sum of all numbers > 1,.
The Question is either wrong, or to be modified,.

It can be modified as,
tan 3 × tan 6 × tan 9 ... tan 87  = 1,.
(or)
tan 0 × tan 3 × tan 6 ...tan 87 × tan 90 = 1
It is possible,.
tan(90 - θ) = cot θ
&
cot θ  = $\frac{1}{tan 0}$
Hence,.
The equation changes to,
⇒ tan 3 × tan 6 × tan 9 ... tan 45 ...tan 87

⇒  cot (90 - 3) × cot (90 - 6) ...1 ...tan 87      (converting tangents in cotangents from 0 to 45)

⇒ cot 87 × cot 84 ...1... tan 87
⇒ $\frac{1}{tan 87} . \frac{1}{tan 84} ... 1..tan 84 .tan87$
⇒ 1
Hence proved

Answer 2

Hey there !!

Answer: 1 .

Step-by-step explanation:

= tan 1 . tan 2 . tan 3 ... tan 44 . tan 45 . tan 46 ... tan 87 . tan 88 . tan 89  

= tan 1 . tan 2 . tan 3 ... tan 44 . tan 45 . tan( 90 - 44 ) ... tan (90 - 3 ) . tan ( 90 - 2 ) . tan ( 90 - 1) 

= tan 1 . tan 2  . tan 3 ...tan44 . tan 45 . cot 44 ... cot 3 . cot 2 . cot 1 

= tan 1 . cot 1 . tan 2 . cot 2 . tan 3 . cot 3 ... tan 44 . cot 44 . tan 45 ... tan 89 . cot 89 

= 1 x 1 x 1 x ... 1 x ... x 1 . [ tan 45 = 1 ] .

= 1 .

Hence, it is solved .

THANKS

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