What is the value of sin(-pi/(12))csc((25pi)/(12)) ?

The value of sin(-pi/(12))csc((25pi)/(12)) is -1

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sin(-pi/(12))csc((25pi)/(12))

sin (- \frac{π}{12}) csc (\frac{25 π}{12})

- 1

Solution steps

sin (- \frac{π}{12}) csc (\frac{25 π}{12})

Use the following property:

sin (- x) = - sin (x)

sin (- \frac{π}{12}) = - sin (\frac{π}{12})

= (- sin (\frac{π}{12})) csc (\frac{25 π}{12})

csc (\frac{25 π}{12}) = csc (\frac{π}{12})

csc (\frac{25 π}{12})

Rewrite

\frac{25 π}{12}

2 π + \frac{π}{12}

= csc (2 π + \frac{π}{12})

Apply the periodicity of

csc

csc (x + 2 π) = csc (x)

csc (2 π + \frac{π}{12}) = csc (\frac{π}{12})

= csc (\frac{π}{12})

= (- sin (\frac{π}{12})) csc (\frac{π}{12})

Simplify

= - sin (\frac{π}{12}) csc (\frac{π}{12})

Rewrite using trig identities:

sin (\frac{π}{12}) = \frac{1}{csc ( \frac{π}{12} )}

sin (\frac{π}{12})

Use the basic trigonometric identity:

sin (x) = \frac{1}{csc ( x )}

= \frac{1}{csc ( \frac{π}{12} )}

= - \frac{1}{csc ( \frac{π}{12} )} csc (\frac{π}{12})

- \frac{1}{csc ( \frac{π}{12} )} csc (\frac{π}{12}) = - 1

- \frac{1}{csc ( \frac{π}{12} )} csc (\frac{π}{12})

Multiply fractions:

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

= - \frac{1 \cdot csc ( \frac{π}{12} )}{csc ( \frac{π}{12} )}

Cancel the common factor:

csc (\frac{π}{12})

= - 1

= - 1

80 \cdot cos (6 0^{\circ})

arcsin (\frac{16}{34})

1 + tan (\frac{π}{4})

3 sin^{2} (π)

15 sin (4 0^{\circ})

What is the value of sin(-pi/(12))csc((25pi)/(12)) ?
The value of sin(-pi/(12))csc((25pi)/(12)) is -1