Is cos((3pi)/2)=cos(pi/2) ?

The answer to whether cos((3pi)/2)=cos(pi/2) is True

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prove cos((3pi)/2)=cos(pi/2)

prove

cos (\frac{3 π}{2}) = cos (\frac{π}{2})

True

Solution steps

cos (\frac{3 π}{2}) = cos (\frac{π}{2})

Manipulating left side

cos (\frac{3 π}{2})

Simplify

cos (\frac{3 π}{2}) : 0

cos (\frac{3 π}{2})

Rewrite using trig identities:

cos (π) cos (\frac{π}{2}) - sin (π) sin (\frac{π}{2})

cos (\frac{3 π}{2})

Write

cos (\frac{3 π}{2})

cos (π + \frac{π}{2})

= cos (π + \frac{π}{2})

Use the Angle Sum identity:

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

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

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

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

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)

Use the following trivial identity:

cos (\frac{π}{2}) = 0

cos (\frac{π}{2})

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}

= 0

= 0

= 0

Manipulating right side

cos (\frac{π}{2})

Simplify

= 0

We showed that the two sides could take the same form

\Rightarrow True

csc (\frac{π}{2} - x) cos (x) = sec (x) cos (x)

1 + 0 + 1 = - 2 cos (0) + 4

(1 - sin^{2} (y)) (1 + tan^{2} (y)) = 1

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

\frac{sin ( x ) + tan ( x )}{1 + sec ( x )} = tan (x)

Is cos((3pi)/2)=cos(pi/2) ?
The answer to whether cos((3pi)/2)=cos(pi/2) is True