What is the value of (tan^2(-(5pi)/6))/(sec(-(5pi)/6)+1) ?

The value of (tan^2(-(5pi)/6))/(sec(-(5pi)/6)+1) is -(2sqrt(3)+3)/3

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(tan^2(-(5pi)/6))/(sec(-(5pi)/6)+1)

\frac{tan ^{2} ( - \frac{5 π}{6} )}{sec ( - \frac{5 π}{6} ) + 1}

- \frac{2 3 + 3}{3}

Solution steps

\frac{tan ^{2} ( - \frac{5 π}{6} )}{sec ( - \frac{5 π}{6} ) + 1}

Use the following property:

sec (- x) = sec (x)

sec (- \frac{5 π}{6}) = sec (\frac{5 π}{6})

= \frac{tan ^{2} ( - \frac{5 π}{6} )}{sec ( \frac{5 π}{6} ) + 1}

Use the following property:

tan (- x) = - tan (x)

tan (- \frac{5 π}{6}) = - tan (\frac{5 π}{6})

= \frac{( - tan ( \frac{5 π}{6} ) ) ^{2}}{sec ( \frac{5 π}{6} ) + 1}

Simplify

= \frac{tan ^{2} ( \frac{5 π}{6} )}{sec ( \frac{5 π}{6} ) + 1}

Rewrite using trig identities:

tan^{2} (\frac{5 π}{6}) = sec^{2} (\frac{5 π}{6}) - 1

= \frac{sec ^{2} ( \frac{5 π}{6} ) - 1}{sec ( \frac{5 π}{6} ) + 1}

Use the following trivial identity:

sec (\frac{5 π}{6}) = - \frac{2 3}{3}

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

Simplify

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

= - \frac{2 3 + 3}{3}

cos (2.5 π)

1 sin (1 0^{\circ})

1 + 2 cos (\frac{π}{6})

\frac{cos ( \frac{11 π}{6} )}{sin ( \frac{11 π}{6} )}

3 + 2 sin (\frac{3 π}{2})

What is the value of (tan^2(-(5pi)/6))/(sec(-(5pi)/6)+1) ?
The value of (tan^2(-(5pi)/6))/(sec(-(5pi)/6)+1) is -(2sqrt(3)+3)/3