Is sec^2(θ/2)= 2/(1+cos(θ)) ?

The answer to whether sec^2(θ/2)= 2/(1+cos(θ)) is True

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prove sec^2(θ/2)= 2/(1+cos(θ))

prove

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

True

Solution steps

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

Let:

u = \frac{θ}{2}

sec^{2} (u) = \frac{2}{1 + cos ( 2 u )}

Prove

sec^{2} (u) = \frac{2}{1 + cos ( 2 u )} :

True

sec^{2} (u) = \frac{2}{1 + cos ( 2 u )}

Manipulating right side

\frac{2}{1 + cos ( 2 u )}

Rewrite using trig identities

\frac{2}{1 + cos ( 2 u )}

Use the Double Angle identity:

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

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

Simplify

\frac{2}{1 + 2 cos ^{2} ( u ) - 1} : \frac{1}{cos ^{2} ( u )}

\frac{2}{1 + 2 cos ^{2} ( u ) - 1}

1 + 2 cos^{2} (u) - 1 = 2 cos^{2} (u)

1 + 2 cos^{2} (u) - 1

Group like terms

= 2 cos^{2} (u) + 1 - 1

1 - 1 = 0

= 2 cos^{2} (u)

= \frac{2}{2 cos ^{2} ( u )}

Divide the numbers:

\frac{2}{2} = 1

= \frac{1}{cos ^{2} ( u )}

= \frac{1}{cos ^{2} ( u )}

= \frac{1}{cos ^{2} ( u )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{( \frac{1}{s e c ( u )} ) ^{2}}

Simplify

\frac{1}{( \frac{1}{s e c ( u )} ) ^{2}}

(\frac{1}{sec ( u )})^{2} = \frac{1}{sec ^{2} ( u )}

(\frac{1}{sec ( u )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{sec ^{2} ( u )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{sec ^{2} ( u )}

= \frac{1}{\frac{1}{s e c ^{2} ( u )}}

Apply the fraction rule:

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

= \frac{sec ^{2} ( u )}{1}

Apply rule

\frac{a}{1} = a

= sec^{2} (u)

sec^{2} (u)

sec^{2} (u)

We showed that the two sides could take the same form

\Rightarrow True

Therefore

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

We showed that the two sides could take the same form

\Rightarrow True

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

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

sin (t) csc (t) = 1

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

cos (x) = sin (x + \frac{π}{2})

Is sec^2(θ/2)= 2/(1+cos(θ)) ?
The answer to whether sec^2(θ/2)= 2/(1+cos(θ)) is True