What is the value of (tan((5pi)/8)-tan((3pi)/8))/(1+tan((5pi)/8)tan((3pi)/8)) ?

The value of (tan((5pi)/8)-tan((3pi)/8))/(1+tan((5pi)/8)tan((3pi)/8)) is 1

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(tan((5pi)/8)-tan((3pi)/8))/(1+tan((5pi)/8)tan((3pi)/8))

\frac{tan ( \frac{5 π}{8} ) - tan ( \frac{3 π}{8} )}{1 + tan ( \frac{5 π}{8} ) tan ( \frac{3 π}{8} )}

1

Solution steps

\frac{tan ( \frac{5 π}{8} ) - tan ( \frac{3 π}{8} )}{1 + tan ( \frac{5 π}{8} ) tan ( \frac{3 π}{8} )}

\frac{tan ( \frac{5 π}{8} ) - tan ( \frac{3 π}{8} )}{1 + tan ( \frac{5 π}{8} ) tan ( \frac{3 π}{8} )} = \frac{tan ( \frac{3 π}{8} ) - tan ( \frac{5 π}{8} )}{- tan ( \frac{5 π}{8} ) tan ( \frac{3 π}{8} ) - 1}

= \frac{tan ( \frac{3 π}{8} ) - tan ( \frac{5 π}{8} )}{- tan ( \frac{5 π}{8} ) tan ( \frac{3 π}{8} ) - 1}

Rewrite using trig identities:

tan (\frac{5 π}{8}) = - tan (\frac{3 π}{8})

= \frac{tan ( \frac{3 π}{8} ) - ( - tan ( \frac{3 π}{8} ) )}{- ( - tan ( \frac{3 π}{8} ) ) tan ( \frac{3 π}{8} ) - 1}

\frac{tan ( \frac{3 π}{8} ) - ( - tan ( \frac{3 π}{8} ) )}{- ( - tan ( \frac{3 π}{8} ) ) tan ( \frac{3 π}{8} ) - 1} = \frac{2 tan ( \frac{3 π}{8} )}{tan ^{2} ( \frac{3 π}{8} ) - 1}

= \frac{2 tan ( \frac{3 π}{8} )}{tan ^{2} ( \frac{3 π}{8} ) - 1}

Rewrite using trig identities:

tan (\frac{3 π}{8}) = 3 + 22

= \frac{2 3 + 2 2}{( 3 + 2 2 ) ^{2} - 1}

Simplify

\frac{2 3 + 2 2}{( 3 + 2 2 ) ^{2} - 1} : 1

= 1

\frac{cot ^{2} ( - \frac{π}{4} )}{1 + 2 sin ( - \frac{π}{4} )}

tan (\frac{15 π}{6})

\frac{15.24}{tan ( 4 1 ^{\circ} )}

arctan (csc (\frac{π}{2}))

arctan (\frac{- 2}{6})

What is the value of (tan((5pi)/8)-tan((3pi)/8))/(1+tan((5pi)/8)tan((3pi)/8)) ?
The value of (tan((5pi)/8)-tan((3pi)/8))/(1+tan((5pi)/8)tan((3pi)/8)) is 1