Is cos^2(0/2)=sec(0) ?

The answer to whether cos^2(0/2)=sec(0) is True

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prove cos^2(0/2)=sec(0)

prove

cos^{2} (\frac{0}{2}) = sec (0)

True

Solution steps

cos^{2} (\frac{0}{2}) = sec (0)

Manipulating left side

cos^{2} (\frac{0}{2})

Simplify

cos^{2} (\frac{0}{2}) : 1

cos^{2} (\frac{0}{2})

Apply rule

\frac{0}{a} = 0, a \neq = 0

\frac{0}{2} = 0

= cos^{2} (0)

Simplify

cos (0) : 1

cos (0)

Use the following trivial identity:

cos (0) = 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^{2}

Apply rule

1^{a} = 1

= 1

= 1

Manipulating right side

sec (0)

Simplify

= 1

We showed that the two sides could take the same form

\Rightarrow True

csc^{2} (x) + 1 = cot^{2} (x) + 2

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

\frac{sin ( x )}{cot ^{2} ( x )} = \frac{tan ^{2} ( x )}{csc ( x )}

sin (x) \cdot (1 + cot (x)) = sin (x) + cos (x)

sin (\frac{5 π}{4}) = cos (\frac{5 π}{4})