What is the value of csc^2((5pi)/4) ?

The value of csc^2((5pi)/4) is 2

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csc^2((5pi)/4)

csc^{2} (\frac{5 π}{4})

2

Solution steps

csc^{2} (\frac{5 π}{4})

Rewrite using trig identities:

1 + cot^{2} (\frac{π}{4})

csc^{2} (\frac{5 π}{4})

Use the Pythagorean identity:

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

= 1 + cot^{2} (\frac{5 π}{4})

cot (\frac{5 π}{4}) = cot (\frac{π}{4})

cot (\frac{5 π}{4})

Rewrite

\frac{5 π}{4}

π + \frac{π}{4}

= cot (π + \frac{π}{4})

Apply the periodicity of

cot

cot (x + π) = cot (x)

cot (π + \frac{π}{4}) = cot (\frac{π}{4})

= cot (\frac{π}{4})

= 1 + cot^{2} (\frac{π}{4})

= 1 + cot^{2} (\frac{π}{4})

Use the following trivial identity:

cot (\frac{π}{4}) = 1

cot (\frac{π}{4})

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

= 1

= 1 + 1^{2}

Simplify

= 2