Cosecant of Angle plus Full Angle

Theorem

$\csc \left({x + 2 \pi}\right) = \csc x$

Proof

\(\ds \csc \left({x + 2 \pi}\right)\)

\(=\)

\(\ds \frac 1 {\sin \left({x + 2 \pi}\right)}\)

Cosecant is Reciprocal of Sine

\(\ds \)

\(=\)

\(\ds \frac 1 {\sin x}\)

Sine of Angle plus Straight Angle

\(\ds \)

\(=\)

\(\ds \csc x\)

Cosecant is Reciprocal of Sine

$\blacksquare$

Also see

• Sine of Angle plus Full Angle
• Cosine of Angle plus Full Angle
• Tangent of Angle plus Full Angle
• Cotangent of Angle plus Full Angle
• Secant of Angle plus Full Angle

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I