Is sec(2θ)=(sec^2(θ)csc^2(θ))/(csc^2(θ)-sec^2(θ)) ?

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

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prove sec(2θ)=(sec^2(θ)csc^2(θ))/(csc^2(θ)-sec^2(θ))

Solution

prove

sec (2 θ) = \frac{sec ^{2} ( θ ) csc ^{2} ( θ )}{csc ^{2} ( θ ) - sec ^{2} ( θ )}

Solution

True

Solution steps

sec (2 θ) = \frac{sec ^{2} ( θ ) csc ^{2} ( θ )}{csc ^{2} ( θ ) - sec ^{2} ( θ )}

Manipulating right side

\frac{sec ^{2} ( θ ) csc ^{2} ( θ )}{csc ^{2} ( θ ) - sec ^{2} ( θ )}

Express with sin, cos

\frac{csc ^{2} ( θ ) sec ^{2} ( θ )}{csc ^{2} ( θ ) - sec ^{2} ( θ )}

Use the basic trigonometric identity:

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

= \frac{( \frac{1}{s i n ( θ )} ) ^{2} sec ^{2} ( θ )}{( \frac{1}{s i n ( θ )} ) ^{2} - sec ^{2} ( θ )}

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

(\frac{1}{sin ( θ )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{sin ^{2} ( θ )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( θ )}

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

(\frac{1}{cos ( θ )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{cos ^{2} ( θ )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( θ )}

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

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

(\frac{1}{sin ( θ )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{sin ^{2} ( θ )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( θ )}

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

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

(\frac{1}{cos ( θ )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{cos ^{2} ( θ )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( θ )}

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

Join

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

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

Least Common Multiplier of

sin^{2} (θ), cos^{2} (θ) : sin^{2} (θ) cos^{2} (θ)

sin^{2} (θ), cos^{2} (θ)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin^{2} (θ)

cos^{2} (θ)

= sin^{2} (θ) cos^{2} (θ)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

sin^{2} (θ) cos^{2} (θ)

For

\frac{1}{sin ^{2} ( θ )} :

multiply the denominator and numerator by

cos^{2} (θ)

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

For

\frac{1}{cos ^{2} ( θ )} :

multiply the denominator and numerator by

sin^{2} (θ)

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

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

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

Multiply

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

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

Refine

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

Cancel the common factor:

sin^{2} (θ)

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

Cancel the common factor:

cos^{2} (θ)

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

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

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

= \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

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Frequently Asked Questions (FAQ)

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