Is csc(pi/2-θ)=sec(θ) ?

The answer to whether csc(pi/2-θ)=sec(θ) is True

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prove csc(pi/2-θ)=sec(θ)

Solution

prove

csc (\frac{π}{2} - θ) = sec (θ)

Solution

True

Solution steps

csc (\frac{π}{2} - θ) = sec (θ)

Manipulating left side

csc (\frac{π}{2} - θ)

Rewrite using trig identities

csc (\frac{π}{2} - θ)

Use the basic trigonometric identity:

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

= \frac{1}{sin ( \frac{π}{2} - θ )}

Use the Angle Difference identity:

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

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

Simplify

\frac{1}{sin ( \frac{π}{2} ) cos ( θ ) - cos ( \frac{π}{2} ) sin ( θ )} : \frac{1}{cos ( θ )}

\frac{1}{sin ( \frac{π}{2} ) cos ( θ ) - cos ( \frac{π}{2} ) sin ( θ )}

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

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

sin (\frac{π}{2}) cos (θ) = cos (θ)

sin (\frac{π}{2}) cos (θ)

Simplify

sin (\frac{π}{2}) : 1

sin (\frac{π}{2})

Use the following trivial identity:

sin (\frac{π}{2}) = 1

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 \cdot cos (θ)

Multiply:

1 \cdot cos (θ) = cos (θ)

= cos (θ)

cos (\frac{π}{2}) sin (θ) = 0

cos (\frac{π}{2}) sin (θ)

Simplify

cos (\frac{π}{2}) : 0

cos (\frac{π}{2})

Use the following trivial identity:

cos (\frac{π}{2}) = 0

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 \cdot sin (θ)

Apply rule

0 \cdot a = 0

= 0

= cos (θ) - 0

cos (θ) - 0 = cos (θ)

= cos (θ)

= \frac{1}{cos ( θ )}

= \frac{1}{cos ( θ )}

= \frac{1}{cos ( θ )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{\frac{1}{s e c ( θ )}}

Simplify

\frac{1}{\frac{1}{s e c ( θ )}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{sec ( θ )}{1}

Apply rule

\frac{a}{1} = a

= sec (θ)

sec (θ)

sec (θ)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is csc(pi/2-θ)=sec(θ) ?
The answer to whether csc(pi/2-θ)=sec(θ) is True