Is (sec^2(a))/(2-sec^2(a))=sec(2a) ?

The answer to whether (sec^2(a))/(2-sec^2(a))=sec(2a) is True

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prove (sec^2(a))/(2-sec^2(a))=sec(2a)

Solution

prove

\frac{sec ^{2} ( a )}{2 - sec ^{2} ( a )} = sec (2 a)

Solution

True

Solution steps

\frac{sec ^{2} ( a )}{2 - sec ^{2} ( a )} = sec (2 a)

Manipulating left side

\frac{sec ^{2} ( a )}{2 - sec ^{2} ( a )}

Express with sin, cos

\frac{sec ^{2} ( a )}{2 - sec ^{2} ( a )}

Use the basic trigonometric identity:

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

= \frac{( \frac{1}{c o s ( a )} ) ^{2}}{2 - ( \frac{1}{c o s ( a )} ) ^{2}}

Simplify

\frac{( \frac{1}{c o s ( a )} ) ^{2}}{2 - ( \frac{1}{c o s ( a )} ) ^{2}} : \frac{1}{2 cos ^{2} ( a ) - 1}

\frac{( \frac{1}{c o s ( a )} ) ^{2}}{2 - ( \frac{1}{c o s ( a )} ) ^{2}}

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

(\frac{1}{cos ( a )})^{2}

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

= \frac{( \frac{1}{c o s ( a )} ) ^{2}}{2 - \frac{1}{c o s ^{2} ( a )}}

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

(\frac{1}{cos ( a )})^{2}

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

= \frac{\frac{1}{c o s ^{2} ( a )}}{2 - \frac{1}{c o s ^{2} ( a )}}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{1}{cos ^{2} ( a ) ( 2 - \frac{1}{c o s ^{2} ( a )} )}

Join

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

2 - \frac{1}{cos ^{2} ( a )}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

= \frac{1}{\frac{2 c o s ^{2} ( a ) - 1}{c o s ^{2} ( a )} cos ^{2} ( a )}

Multiply

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

cos^{2} (a) \frac{2 cos ^{2} ( a ) - 1}{cos ^{2} ( a )}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Cancel the common factor:

cos^{2} (a)

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

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

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

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

= \frac{1}{cos ( 2 a )}

= \frac{1}{cos ( 2 a )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{\frac{1}{s e c ( 2 a )}}

Simplify

\frac{1}{\frac{1}{s e c ( 2 a )}}

Apply the fraction rule:

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

= \frac{sec ( 2 a )}{1}

Apply rule

\frac{a}{1} = a

= sec (2 a)

sec (2 a)

sec (2 a)

We showed that the two sides could take the same form

\Rightarrow True

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Is (sec^2(a))/(2-sec^2(a))=sec(2a) ?
The answer to whether (sec^2(a))/(2-sec^2(a))=sec(2a) is True