Question

$\tan(a) = \frac{ \sin(17) - \cos(17) }{ \cos(17 ) + \sin(17) } \\ \\ find \: the \: value \: of \: \tan(a)$
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Answer 1

Solution -

Let [ sin 17 - cos 17 ]/[ sin 17 + cos 17 ] = tan a = ß .

Let sin 17 = δ

Then , cos 17 = √(1- δ² )

The given equation now becomes -

> [ δ - √(1- δ²) ]/[ δ + √(1- δ²) ] = ß

Applying componendo and dividendo ;

> (ß+1)/(ß-1) = - δ/[ √( 1 - δ² ) ]

Substituting the values of δ and √(1- δ²)

> (ß + 1)/( ß - 1) = sin 17/ cos 17

> (ß + 1)/(ẞ - 1) = tan 17

Applying componendo dividendo again , both sides !

> [ ß + 1 + ß - 1 ]/[ ß + 1 - ß + 1] = [ tan 17 + 1]/[ tan 17 - 1 ]

> ß = [ tan 17 + 1 ]/[ tan 17 - 1 ]

> ß = - [ 1 + tan 17 ]/[ 1 - tan 17]

> ß = - [ tan ( 45 + 17) ]

> ß = - tan 62 = tan 28°

Hence

tan ( a ) = tan ( 28°)

> a = 28° .

This is the required answer !

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Answer 2

*Answer:-

Let x = ( cos17 + sin17 ) / ( cos17 − sin17 )

Dividing Numerator & Denominator by cos 17

= ( 1 + tan17 ) / ( 1 − tan17 )

We know tan 45 = 1,

= ( tan45 + tan17 ) / ( 1 − tan45.tan17 )

tan( A + B ) = tan A + tan B / 1 - tan A.tan B

= tan62

Thus, ( cos 17 +sin 17 )/(cos 17-sin17) = tan 62