Is tan(180-y)=-tan(y) ?

The answer to whether tan(180-y)=-tan(y) is True

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prove tan(180-y)=-tan(y)

Solution

prove

tan (18 0^{\circ} - y) = - tan (y)

Solution

True

Solution steps

tan (18 0^{\circ} - y) = - tan (y)

Manipulating left side

tan (18 0^{\circ} - y)

Rewrite using trig identities

tan (18 0^{\circ} - y)

Use the basic trigonometric identity:

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

= \frac{sin ( 18 0 ^{\circ} - y )}{cos ( 18 0 ^{\circ} - y )}

Use the Angle Difference identity:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= \frac{sin ( 18 0 ^{\circ} ) cos ( y ) - cos ( 18 0 ^{\circ} ) sin ( y )}{cos ( 18 0 ^{\circ} - y )}

Use the Angle Difference identity:

cos (s - t) = cos (s) cos (t) + sin (s) sin (t)

= \frac{sin ( 18 0 ^{\circ} ) cos ( y ) - cos ( 18 0 ^{\circ} ) sin ( y )}{cos ( 18 0 ^{\circ} ) cos ( y ) + sin ( 18 0 ^{\circ} ) sin ( y )}

Simplify

\frac{sin ( 18 0 ^{\circ} ) cos ( y ) - cos ( 18 0 ^{\circ} ) sin ( y )}{cos ( 18 0 ^{\circ} ) cos ( y ) + sin ( 18 0 ^{\circ} ) sin ( y )} : - \frac{sin ( y )}{cos ( y )}

\frac{sin ( 18 0 ^{\circ} ) cos ( y ) - cos ( 18 0 ^{\circ} ) sin ( y )}{cos ( 18 0 ^{\circ} ) cos ( y ) + sin ( 18 0 ^{\circ} ) sin ( y )}

sin (18 0^{\circ}) cos (y) - cos (18 0^{\circ}) sin (y) = sin (y)

sin (18 0^{\circ}) cos (y) - cos (18 0^{\circ}) sin (y)

sin (18 0^{\circ}) cos (y) = 0

sin (18 0^{\circ}) cos (y)

Simplify

sin (18 0^{\circ}) : 0

sin (18 0^{\circ})

Use the following trivial identity:

sin (18 0^{\circ}) = 0

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

= 0 \cdot cos (y)

Apply rule

0 \cdot a = 0

= 0

cos (18 0^{\circ}) sin (y) = - sin (y)

cos (18 0^{\circ}) sin (y)

Simplify

cos (18 0^{\circ}) : - 1

cos (18 0^{\circ})

Use the following trivial identity:

cos (18 0^{\circ}) = (- 1)

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - 1

= - 1 \cdot sin (y)

Multiply:

1 \cdot sin (y) = sin (y)

= - sin (y)

= 0 - (- sin (y))

Refine

= sin (y)

= \frac{sin ( y )}{cos ( 18 0 ^{\circ} ) cos ( y ) + sin ( 18 0 ^{\circ} ) sin ( y )}

cos (18 0^{\circ}) cos (y) + sin (18 0^{\circ}) sin (y) = - cos (y)

cos (18 0^{\circ}) cos (y) + sin (18 0^{\circ}) sin (y)

cos (18 0^{\circ}) cos (y) = - cos (y)

cos (18 0^{\circ}) cos (y)

Simplify

cos (18 0^{\circ}) : - 1

cos (18 0^{\circ})

Use the following trivial identity:

cos (18 0^{\circ}) = (- 1)

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - 1

= - 1 \cdot cos (y)

Multiply:

1 \cdot cos (y) = cos (y)

= - cos (y)

= - cos (y) + sin (18 0^{\circ}) sin (y)

sin (18 0^{\circ}) sin (y) = 0

sin (18 0^{\circ}) sin (y)

Simplify

sin (18 0^{\circ}) : 0

sin (18 0^{\circ})

Use the following trivial identity:

sin (18 0^{\circ}) = 0

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

= 0 \cdot sin (y)

Apply rule

0 \cdot a = 0

= 0

= - cos (y) + 0

- cos (y) + 0 = - cos (y)

= - cos (y)

= \frac{sin ( y )}{- cos ( y )}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{sin ( y )}{cos ( y )}

= - \frac{sin ( y )}{cos ( y )}

= - \frac{sin ( y )}{cos ( y )}

Use the basic trigonometric identity:

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

= - tan (y)

We showed that the two sides could take the same form

\Rightarrow True

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Is tan(180-y)=-tan(y) ?
The answer to whether tan(180-y)=-tan(y) is True