What is the value of -csc^2((7pi)/6) ?

The value of -csc^2((7pi)/6) is -4

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-csc^2((7pi)/6)

- csc^{2} (\frac{7 π}{6})

- 4

Solution steps

- csc^{2} (\frac{7 π}{6})

Rewrite using trig identities:

csc^{2} (\frac{7 π}{6}) = 1 + cot^{2} (\frac{π}{6})

csc^{2} (\frac{7 π}{6})

Use the Pythagorean identity:

csc^{2} (x) = 1 + cot^{2} (x)

= 1 + cot^{2} (\frac{7 π}{6})

cot (\frac{7 π}{6}) = cot (\frac{π}{6})

cot (\frac{7 π}{6})

Rewrite

\frac{7 π}{6}

π + \frac{π}{6}

= cot (π + \frac{π}{6})

Apply the periodicity of

cot

cot (x + π) = cot (x)

cot (π + \frac{π}{6}) = cot (\frac{π}{6})

= cot (\frac{π}{6})

= 1 + cot^{2} (\frac{π}{6})

= - (1 + cot^{2} (\frac{π}{6}))

Simplify

= - 1 - cot^{2} (\frac{π}{6})

Use the following trivial identity:

cot (\frac{π}{6}) = 3

cot (\frac{π}{6})

cot (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

= 3

= - 1 - (3)^{2}

Simplify

- 1 - (3)^{2} : - 4

- 1 - (3)^{2}

(3)^{2} = 3

(3)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 3

= - 1 - 3

Subtract the numbers:

- 1 - 3 = - 4

= - 4

= - 4

sin^{2} (6 0^{\circ}) + tan^{2} (6 0^{\circ}) - csc^{2} (3 0^{\circ})

(875)^{3} - sin (87 5^{\circ})

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

cos (arccos (\frac{π}{2}))

tan (- 40 5^{\circ})