Is cos(x)(sec^2(x))=sec(x) ?

The answer to whether cos(x)(sec^2(x))=sec(x) is True

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prove cos(x)(sec^2(x))=sec(x)

prove

cos (x) (sec^{2} (x)) = sec (x)

True

Solution steps

cos (x) sec^{2} (x) = sec (x)

Manipulating left side

cos (x) sec^{2} (x)

Express with sin, cos

cos (x) sec^{2} (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Multiply:

1 \cdot cos (x) = cos (x)

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

Cancel the common factor:

cos (x)

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{\frac{1}{s e c ( x )}}

Simplify

\frac{1}{\frac{1}{s e c ( x )}}

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= sec (x)

sec (x)

sec (x)

We showed that the two sides could take the same form

\Rightarrow True

csc (x) + cot^{2} (x) csc (x) = csc^{3} (x)

cot (x) = cos (x) arcsin (x)

\frac{1}{sin ( x ) sec ( x )} = cot (x)

3 + sin^{2} (x) = 4 - cos^{2} (x)

sin^{4} (m) - cos^{4} (m) = 1 - 2 cos^{2} (m)

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