Is sin(pi/2-θ)=sin(θ-pi/2) ?

The answer to whether sin(pi/2-θ)=sin(θ-pi/2) is False

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prove sin(pi/2-θ)=sin(θ-pi/2)

prove

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

False

Solution steps

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

Show that the two sides are not equal

For

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

plug in

θ = 0

sin (\frac{π}{2} - 0) = 1

sin (\frac{π}{2} - 0)

Simplify

= sin (\frac{π}{2})

Use the following trivial identity:

sin (\frac{π}{2}) = 1

sin (\frac{π}{2})

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}

= 1

= 1

sin (0 - \frac{π}{2}) = - 1

sin (0 - \frac{π}{2})

Simplify

= sin (- \frac{π}{2})

Use the following property:

sin (- x) = - sin (x)

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

= - sin (\frac{π}{2})

Use the following trivial identity:

sin (\frac{π}{2}) = 1

sin (\frac{π}{2})

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}

= 1

= - 1

The two sides are not equal

\Rightarrow False

cos (- 2 x) = cos (2 x)

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

csc (x) - cot (x) cos (x) = \frac{1}{csc ( x )}

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

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

Is sin(pi/2-θ)=sin(θ-pi/2) ?
The answer to whether sin(pi/2-θ)=sin(θ-pi/2) is False