What is the value of tan((5pi)/6+(7pi)/4) ?

The value of tan((5pi)/6+(7pi)/4) is -2-sqrt(3)

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tan((5pi)/6+(7pi)/4)

tan (\frac{5 π}{6} + \frac{7 π}{4})

- 2 - 3

Solution steps

tan (\frac{5 π}{6} + \frac{7 π}{4})

Simplify:

\frac{5 π}{6} + \frac{7 π}{4} = \frac{31 π}{12}

= tan (\frac{31 π}{12})

tan (\frac{31 π}{12}) = tan (\frac{7 π}{12})

= tan (\frac{7 π}{12})

Rewrite using trig identities:

\frac{tan ( \frac{π}{3} ) + tan ( \frac{π}{4} )}{1 - tan ( \frac{π}{3} ) tan ( \frac{π}{4} )}

= \frac{tan ( \frac{π}{3} ) + tan ( \frac{π}{4} )}{1 - tan ( \frac{π}{3} ) tan ( \frac{π}{4} )}

Use the following trivial identity:

tan (\frac{π}{3}) = 3

Use the following trivial identity:

tan (\frac{π}{4}) = 1

= \frac{3 + 1}{1 - 3 \cdot 1}

Simplify

\frac{3 + 1}{1 - 3 \cdot 1} : - 2 - 3

= - 2 - 3

2 sec (\frac{π}{3})

tan (2 arccos (\frac{4}{5}))

sec (102 0^{\circ})

- 2 cos (\frac{5 π}{3})

cot (8 1^{\circ})

What is the value of tan((5pi)/6+(7pi)/4) ?
The value of tan((5pi)/6+(7pi)/4) is -2-sqrt(3)