Question

what is the value of cot 180/8 and tan 180/8?

Answer 1

Hi ,

We know that
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Cot 2θ = ( cot^2 θ - 1 ) / 2cot θ
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i ) Let θ = 180 / 8 = 22.5

Cot θ = a

Then ,

Cot 2θ = ( cot^2θ - 1 )/ 2 cotθ

Cot ( 2 × 22.5 ) = ( a^2 - 1 )/2a

Cot 45 = ( a^2 - 1 )/ 2a

1 = ( a^2 - 1)/2a

2a = a^2 - 1

a^2 -2a = 1

add 1 bothsides

a^2 -2a +1 = 2

( a - 1 )^2 = 2

a - 1 = sqrt 2

we are taking positive

value because cot 22.5 lies in 1st

Quadrant.

Therefore ,

a = sqrt 2 + 1

Cot θ = sqrt2 + 1

Cot 180 / 8 = sqrt 2 + 1 ------( 1 )

ii ) tan 180 / 8

= 1 / ( cot 180 / 8 )

= 1 / ( sqrt 2 + 1 ) from ( 1 )

= ( sqrt 2 - 1 )/ ( sqrt 2 +1)(sqrt2-1)

= ( sqrt 2 - 1 )/ [ ( sqrt2 )^2 - 1 ^2 ]

= ( sqrt 2 - 1 )/ ( 2 - 1 )

= sqrt 2 - 1

Therefore ,

Tan 180/8 = sqrt2 - 1

I hope this helps you.

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