Is sec^2(t)=cos^{-2}(t) ?

The answer to whether sec^2(t)=cos^{-2}(t) is True

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prove sec^2(t)=cos^{-2}(t)

prove

sec^{2} (t) = cos^{- 2} (t)

True

Solution steps

sec^{2} (t) = cos^{- 2} (t)

Manipulating right side

cos^{- 2} (t)

Simplify

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= sec^{2} (t)

sec^{2} (t)

sec^{2} (t)

We showed that the two sides could take the same form

\Rightarrow True

\frac{tan ( x ) \cdot cot ( x )}{csc ( x )} = sin (x)

\frac{1}{sin ^{2} ( x ) tan ( x )} = csc^{2} (x) cot (x)

sec (x) - cos (x) = (sin (x)) (tan (x))

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

\frac{1}{1 + cot ^{2} ( a )} = sin^{2} (a)