Is 1/(sin(x))=arcsin(x) ?

The answer to whether 1/(sin(x))=arcsin(x) is False

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prove 1/(sin(x))=arcsin(x)

prove

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

False

Solution steps

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

Show that the two sides are not equal

For

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

plug in

x = 1

\frac{1}{sin ( 1 )} = 1.18839 \dots

\frac{1}{sin ( 1 )}

Refine to a decimal form

= 1.18839 \dots

arcsin (1) = \frac{π}{2} (Decimal : Degrees : 1.57079 \dots 9 0^{\circ})

arcsin (1)

Use the following trivial identity:

arcsin (1) = \frac{π}{2}

arcsin (1)

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{2}

= \frac{π}{2}

The two sides are not equal

\Rightarrow False

\frac{1 - 2 sin ^{2} ( x )}{1 - sin ^{2} ( x )} = 1 - tan (2 x)

cot (x) = arctan (x)

\frac{1}{sin ( x ) cos ( x )} = \frac{2}{sin ( 2 x )}

sin (x) tan (x) = 1 + cos (x)

(\frac{1 + sin ( x )}{cos ( x )})^{2} + (\frac{1 - sin ( x )}{cos ( x )})^{2} = 2 + 4 tan^{2} (x)