Is csc(2y)=(csc(y))/(2cos(y)) ?

The answer to whether csc(2y)=(csc(y))/(2cos(y)) is True

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prove csc(2y)=(csc(y))/(2cos(y))

prove

csc (2 y) = \frac{csc ( y )}{2 cos ( y )}

True

Solution steps

csc (2 y) = \frac{csc ( y )}{2 cos ( y )}

Manipulating left side

csc (2 y)

Express with sin, cos

csc (2 y)

Use the basic trigonometric identity:

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

= \frac{1}{sin ( 2 y )}

= \frac{1}{sin ( 2 y )}

Rewrite using trig identities

\frac{1}{sin ( 2 y )}

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{2 \cdot \frac{1}{c s c ( y )} cos ( y )}

Simplify

\frac{1}{2 \cdot \frac{1}{c s c ( y )} cos ( y )}

Multiply

2 \cdot \frac{1}{csc ( y )} cos (y) : \frac{2 cos ( y )}{csc ( y )}

2 \cdot \frac{1}{csc ( y )} cos (y)

Multiply fractions:

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

= \frac{1 \cdot 2 cos ( y )}{csc ( y )}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 cos ( y )}{csc ( y )}

= \frac{1}{\frac{2 c o s ( y )}{c s c ( y )}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{csc ( y )}{2 cos ( y )}

\frac{csc ( y )}{2 cos ( y )}

\frac{csc ( y )}{2 cos ( y )}

We showed that the two sides could take the same form

\Rightarrow True

cos (θ - 27 0^{\circ}) = - sin (θ)

1 + tan^{2} (x) + cot (x) = sec^{2} (x) + cos (x)

cos (θ) sec (θ) = tan (θ) cot (θ)

cos (x) - cos (x) = 2 cos (x)

\frac{cos ^{2} ( 2 θ )}{2} = \frac{1 + cos ( 4 θ )}{8}

Is csc(2y)=(csc(y))/(2cos(y)) ?
The answer to whether csc(2y)=(csc(y))/(2cos(y)) is True