Is sec(2θ)=(1+tan^2(θ))/(1-tan^2(θ)) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove sec(2θ)=(1+tan^2(θ))/(1-tan^2(θ))

Solution

prove

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

Solution

True

Solution steps

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

Manipulating right side

\frac{1 + tan ^{2} ( θ )}{1 - tan ^{2} ( θ )}

Express with sin, cos

\frac{1 + tan ^{2} ( θ )}{1 - tan ^{2} ( θ )}

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{1 + ( \frac{s i n ( θ )}{c o s ( θ )} ) ^{2}}{1 - ( \frac{s i n ( θ )}{c o s ( θ )} ) ^{2}}

Simplify

\frac{1 + ( \frac{s i n ( θ )}{c o s ( θ )} ) ^{2}}{1 - ( \frac{s i n ( θ )}{c o s ( θ )} ) ^{2}} : \frac{cos ^{2} ( θ ) + sin ^{2} ( θ )}{cos ^{2} ( θ ) - sin ^{2} ( θ )}

\frac{1 + ( \frac{s i n ( θ )}{c o s ( θ )} ) ^{2}}{1 - ( \frac{s i n ( θ )}{c o s ( θ )} ) ^{2}}

Apply exponent rule:

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

= \frac{1 + ( \frac{s i n ( θ )}{c o s ( θ )} ) ^{2}}{1 - \frac{s i n ^{2} ( θ )}{c o s ^{2} ( θ )}}

Apply exponent rule:

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

= \frac{1 + \frac{s i n ^{2} ( θ )}{c o s ^{2} ( θ )}}{1 - \frac{s i n ^{2} ( θ )}{c o s ^{2} ( θ )}}

Join

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot cos^{2} (θ) = cos^{2} (θ)

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

= \frac{1 + \frac{s i n ^{2} ( θ )}{c o s ^{2} ( θ )}}{\frac{c o s ^{2} ( θ ) - s i n ^{2} ( θ )}{c o s ^{2} ( θ )}}

Join

1 + \frac{sin ^{2} ( θ )}{cos ^{2} ( θ )} : \frac{cos ^{2} ( θ ) + sin ^{2} ( θ )}{cos ^{2} ( θ )}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot cos^{2} (θ) = cos^{2} (θ)

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

= \frac{\frac{c o s ^{2} ( θ ) + s i n ^{2} ( θ )}{c o s ^{2} ( θ )}}{\frac{c o s ^{2} ( θ ) - s i n ^{2} ( θ )}{c o s ^{2} ( θ )}}

Divide fractions:

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

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

Cancel the common factor:

cos^{2} (θ)

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

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

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

Rewrite using trig identities

\frac{cos ^{2} ( θ ) + sin ^{2} ( θ )}{cos ^{2} ( θ ) - sin ^{2} ( θ )}

Use the Double Angle identity:

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

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

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

= \frac{1}{cos ( 2 θ )}

= \frac{1}{cos ( 2 θ )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{\frac{1}{s e c ( 2 θ )}}

Simplify

\frac{1}{\frac{1}{s e c ( 2 θ )}}

Apply the fraction rule:

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

= \frac{sec ( 2 θ )}{1}

Apply rule

\frac{a}{1} = a

= sec (2 θ)

sec (2 θ)

sec (2 θ)

We showed that the two sides could take the same form

\Rightarrow True

Popular Examples

prove tan(x-x)=-tan(x)prove

tan (x - x) = - tan (x)

prove (1-(sin(x)+cos(x))^2)/(sin(2x))=-1 prove

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

prove ((cos(x)+1))/(sin^3(x))=((csc(x)))/(1-cos(x))prove

\frac{( cos ( x ) + 1 )}{sin ^{3} ( x )} = \frac{( csc ( x ) )}{1 - cos ( x )}

prove cos(x)=(sin(2x))/(2sin(x))prove

cos (x) = \frac{sin ( 2 x )}{2 sin ( x )}

prove (1+cot^2(x))/(sec^2(x))=cot^2(x)prove

\frac{1 + cot ^{2} ( x )}{sec ^{2} ( x )} = cot^{2} (x)

Frequently Asked Questions (FAQ)

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