Is 1-cos^2(x)=(tan^2(x))/(sec^2(x)) ?

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

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prove 1-cos^2(x)=(tan^2(x))/(sec^2(x))

prove

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

True

Solution steps

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

Manipulating right side

\frac{tan ^{2} ( x )}{sec ^{2} ( x )}

Express with sin, cos

\frac{tan ^{2} ( x )}{sec ^{2} ( x )}

Use the basic trigonometric identity:

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

= \frac{( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{sec ^{2} ( x )}

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

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

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

Apply exponent rule:

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

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

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

Divide fractions:

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

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

Refine

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

Cancel the common factor:

cos^{2} (x)

= sin^{2} (x)

= sin^{2} (x)

= sin^{2} (x)

Rewrite using trig identities

sin^{2} (x)

Use the Pythagorean identity:

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

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

= 1 - cos^{2} (x)

= 1 - cos^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

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