Is 2cos(x)*csc^2(x)=2cot(x)*csc(x) ?

The answer to whether 2cos(x)*csc^2(x)=2cot(x)*csc(x) is True

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prove 2cos(x)csc^2(x)=2cot(x)csc(x)

Solution

prove

2 cos (x) \cdot csc^{2} (x) = 2 cot (x) \cdot csc (x)

Solution

True

Solution steps

2 cos (x) csc^{2} (x) = 2 cot (x) csc (x)

Manipulating right side

2 cot (x) csc (x)

Express with sin, cos

2 cot (x) csc (x)

Use the basic trigonometric identity:

cot (x) = \frac{cos ( x )}{sin ( x )}

= 2 \cdot \frac{cos ( x )}{sin ( x )} csc (x)

Use the basic trigonometric identity:

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

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

Simplify

2 \cdot \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )} : \frac{2 cos ( x )}{sin ^{2} ( x )}

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin (x) = sin^{1 + 1} (x)

= sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= sin^{2} (x)

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{2 cos ( x )}{( \frac{1}{c s c ( x )} ) ^{2}}

Simplify

\frac{2 cos ( x )}{( \frac{1}{c s c ( x )} ) ^{2}}

(\frac{1}{csc ( x )})^{2} = \frac{1}{csc ^{2} ( x )}

(\frac{1}{csc ( x )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{csc ^{2} ( x )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{csc ^{2} ( x )}

= \frac{2 cos ( x )}{\frac{1}{c s c ^{2} ( x )}}

Apply the fraction rule:

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

= \frac{2 cos ( x ) csc ^{2} ( x )}{1}

Apply rule

\frac{a}{1} = a

= 2 csc^{2} (x) cos (x)

2 csc^{2} (x) cos (x)

2 csc^{2} (x) cos (x)

= 2 cos (x) csc^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is 2cos(x)*csc^2(x)=2cot(x)*csc(x) ?
The answer to whether 2cos(x)*csc^2(x)=2cot(x)*csc(x) is True

prove 2cos(x)*csc^2(x)=2cot(x)*csc(x)

Solution

Solution

Popular Examples

Frequently Asked Questions (FAQ)

Is 2cos(x)*csc^2(x)=2cot(x)*csc(x) ?

prove 2cos(x)csc^2(x)=2cot(x)csc(x)

Is 2cos(x)csc^2(x)=2cot(x)csc(x) ?