Is tan((7u)/2)=csc(7u)-cot(7u) ?

The answer to whether tan((7u)/2)=csc(7u)-cot(7u) is True

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prove tan((7u)/2)=csc(7u)-cot(7u)

Solution

prove

tan (\frac{7 u}{2}) = csc (7 u) - cot (7 u)

Solution

True

Solution steps

tan (\frac{7 u}{2}) = csc (7 u) - cot (7 u)

Let:

v = \frac{7 u}{2}

tan (v) = csc (2 v) - cot (2 v)

Prove

tan (v) = csc (2 v) - cot (2 v) :

True

tan (v) = csc (2 v) - cot (2 v)

Manipulating right side

csc (2 v) - cot (2 v)

Express with sin, cos

- cot (2 v) + csc (2 v)

Use the basic trigonometric identity:

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

= - \frac{cos ( 2 v )}{sin ( 2 v )} + csc (2 v)

Use the basic trigonometric identity:

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

= - \frac{cos ( 2 v )}{sin ( 2 v )} + \frac{1}{sin ( 2 v )}

Simplify

- \frac{cos ( 2 v )}{sin ( 2 v )} + \frac{1}{sin ( 2 v )} : \frac{- cos ( 2 v ) + 1}{sin ( 2 v )}

- \frac{cos ( 2 v )}{sin ( 2 v )} + \frac{1}{sin ( 2 v )}

Apply rule

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

= \frac{- cos ( 2 v ) + 1}{sin ( 2 v )}

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

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

Rewrite using trig identities

\frac{1 - cos ( 2 v )}{sin ( 2 v )}

Use the Double Angle identity:

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

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

Use the Double Angle identity:

cos (2 x) = 1 - 2 sin^{2} (x)

= \frac{1 - ( 1 - 2 sin ^{2} ( v ) )}{2 sin ( v ) cos ( v )}

Simplify

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

\frac{1 - ( 1 - 2 sin ^{2} ( v ) )}{2 sin ( v ) cos ( v )}

Expand

1 - (1 - 2 sin^{2} (v)) : 2 sin^{2} (v)

1 - (1 - 2 sin^{2} (v))

- (1 - 2 sin^{2} (v)) : - 1 + 2 sin^{2} (v)

- (1 - 2 sin^{2} (v))

Distribute parentheses

= - (1) - (- 2 sin^{2} (v))

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 1 + 2 sin^{2} (v)

= 1 - 1 + 2 sin^{2} (v)

1 - 1 = 0

= 2 sin^{2} (v)

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

Divide the numbers:

\frac{2}{2} = 1

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

Cancel the common factor:

sin (v)

= \frac{sin ( v )}{cos ( v )}

= \frac{sin ( v )}{cos ( v )}

= \frac{sin ( v )}{cos ( v )}

Use the basic trigonometric identity:

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

= tan (v)

We showed that the two sides could take the same form

\Rightarrow True

Therefore

tan (\frac{7 u}{2}) = csc (7 u) - cot (7 u)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is tan((7u)/2)=csc(7u)-cot(7u) ?
The answer to whether tan((7u)/2)=csc(7u)-cot(7u) is True