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

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

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

Solution

prove

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

Solution

True

Solution steps

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

Manipulating left side

\frac{tan ^{2} ( t ) - 1}{sec ^{2} ( t )}

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Apply exponent rule:

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

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

Apply the fraction rule:

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

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

Join

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

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

Convert element to fraction:

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

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

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} ( t ) + sin ^{2} ( t )}{cos ^{2} ( t )}

Multiply:

1 \cdot cos^{2} (t) = cos^{2} (t)

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

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

Apply the fraction rule:

\frac{a}{1} = a

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

Multiply fractions:

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

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

Cancel the common factor:

cos^{2} (t)

= - - cos^{2} (t) + sin^{2} (t)

= - cos^{2} (t) + sin^{2} (t)

= - cos^{2} (t) + sin^{2} (t)

Manipulating right side

\frac{tan ( t ) - cot ( t )}{tan ( t ) + cot ( t )}

Express with sin, cos

\frac{- cot ( t ) + tan ( t )}{cot ( t ) + tan ( t )}

Use the basic trigonometric identity:

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

= \frac{- \frac{c o s ( t )}{s i n ( t )} + tan ( t )}{\frac{c o s ( t )}{s i n ( t )} + tan ( t )}

Use the basic trigonometric identity:

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

= \frac{- \frac{c o s ( t )}{s i n ( t )} + \frac{s i n ( t )}{c o s ( t )}}{\frac{c o s ( t )}{s i n ( t )} + \frac{s i n ( t )}{c o s ( t )}}

Simplify

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

\frac{- \frac{c o s ( t )}{s i n ( t )} + \frac{s i n ( t )}{c o s ( t )}}{\frac{c o s ( t )}{s i n ( t )} + \frac{s i n ( t )}{c o s ( t )}}

Join

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

\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}

Least Common Multiplier of

sin (t), cos (t) : sin (t) cos (t)

sin (t), cos (t)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (t)

cos (t)

= sin (t) cos (t)

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 (t) cos (t)

For

\frac{cos ( t )}{sin ( t )} :

multiply the denominator and numerator by

cos (t)

\frac{cos ( t )}{sin ( t )} = \frac{cos ( t ) cos ( t )}{sin ( t ) cos ( t )} = \frac{cos ^{2} ( t )}{sin ( t ) cos ( t )}

For

\frac{sin ( t )}{cos ( t )} :

multiply the denominator and numerator by

sin (t)

\frac{sin ( t )}{cos ( t )} = \frac{sin ( t ) sin ( t )}{cos ( t ) sin ( t )} = \frac{sin ^{2} ( t )}{sin ( t ) cos ( t )}

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

Since the denominators are equal, combine the fractions:

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

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

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

Join

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

- \frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}

Least Common Multiplier of

sin (t), cos (t) : sin (t) cos (t)

sin (t), cos (t)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (t)

cos (t)

= sin (t) cos (t)

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 (t) cos (t)

For

\frac{cos ( t )}{sin ( t )} :

multiply the denominator and numerator by

cos (t)

\frac{cos ( t )}{sin ( t )} = \frac{cos ( t ) cos ( t )}{sin ( t ) cos ( t )} = \frac{cos ^{2} ( t )}{sin ( t ) cos ( t )}

For

\frac{sin ( t )}{cos ( t )} :

multiply the denominator and numerator by

sin (t)

\frac{sin ( t )}{cos ( t )} = \frac{sin ( t ) sin ( t )}{cos ( t ) sin ( t )} = \frac{sin ^{2} ( t )}{sin ( t ) cos ( t )}

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

Since the denominators are equal, combine the fractions:

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

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

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

Divide fractions:

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

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

Cancel the common factor:

sin (t)

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

Cancel the common factor:

cos (t)

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

Apply rule

\frac{a}{1} = a

= - cos^{2} (t) + sin^{2} (t)

= - cos^{2} (t) + sin^{2} (t)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

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