Is sin(x+pi)=-sin(x) ?

The answer to whether sin(x+pi)=-sin(x) is True

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prove sin(x+pi)=-sin(x)

prove

sin (x + π) = - sin (x)

True

Solution steps

sin (x + π) = - sin (x)

Manipulating left side

sin (x + π)

Rewrite using trig identities

sin (x + π)

Use the Angle Sum identity:

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

= sin (x) cos (π) + cos (x) sin (π)

Simplify

sin (x) cos (π) + cos (x) sin (π) : - sin (x)

sin (x) cos (π) + cos (x) sin (π)

sin (x) cos (π) = - sin (x)

sin (x) cos (π)

Simplify

cos (π) : - 1

cos (π)

Use the following trivial identity:

cos (π) = (- 1)

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - 1

= (- 1) sin (x)

Refine

= - sin (x)

= - sin (x) + sin (π) cos (x)

cos (x) sin (π) = 0

cos (x) sin (π)

Simplify

sin (π) : 0

sin (π)

Use the following trivial identity:

sin (π) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} 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 (x)

Apply rule

0 \cdot a = 0

= 0

= - sin (x) + 0

- sin (x) + 0 = - sin (x)

= - sin (x)

= - sin (x)

= - sin (x)

We showed that the two sides could take the same form

\Rightarrow True

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

(cot^{2} (x) + 1) (sin^{2} (x) - 1) = - cot^{2} (x)

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

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

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