Is 1/2 cot(x)-1/2 tan(x)=cot(2x) ?

The answer to whether 1/2 cot(x)-1/2 tan(x)=cot(2x) is True

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prove 1/2 cot(x)-1/2 tan(x)=cot(2x)

Solution

prove

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

Solution

True

Solution steps

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

Manipulating left side

\frac{1}{2} cot (x) - \frac{1}{2} tan (x)

Express with sin, cos

cot (x) \frac{1}{2} - \frac{1}{2} tan (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Multiply fractions:

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

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

Multiply:

cos (x) \cdot 1 = cos (x)

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

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

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

Multiply fractions:

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

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

Multiply:

1 \cdot sin (x) = sin (x)

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

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

Least Common Multiplier of

sin (x) 2, 2 cos (x) : 2 sin (x) cos (x)

sin (x) \cdot 2, 2 cos (x)

Lowest Common Multiplier (LCM)

Least Common Multiplier of

2, 2 : 2

2, 2

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

2

2

= 2

Multiply the numbers:

2 = 2

= 2

Compute an expression comprised of factors that appear either in

sin (x) 2

2 cos (x)

= 2 sin (x) cos (x)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

2 sin (x) cos (x)

For

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

multiply the denominator and numerator by

cos (x)

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

For

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

multiply the denominator and numerator by

sin (x)

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Use the Double Angle identity:

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

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

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

Use the basic trigonometric identity:

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

= cot (2 x)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is 1/2 cot(x)-1/2 tan(x)=cot(2x) ?
The answer to whether 1/2 cot(x)-1/2 tan(x)=cot(2x) is True