Is tan(x)sec^4(x)=(sin(x))/(cos^5(x)) ?

The answer to whether tan(x)sec^4(x)=(sin(x))/(cos^5(x)) is True

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prove tan(x)sec^4(x)=(sin(x))/(cos^5(x))

prove

tan (x) sec^{4} (x) = \frac{sin ( x )}{cos ^{5} ( x )}

True

Solution steps

tan (x) sec^{4} (x) = \frac{sin ( x )}{cos ^{5} ( x )}

Manipulating left side

tan (x) sec^{4} (x)

Express with sin, cos

sec^{4} (x) tan (x)

Use the basic trigonometric identity:

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

= (\frac{1}{cos ( x )})^{4} tan (x)

Use the basic trigonometric identity:

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

= (\frac{1}{cos ( x )})^{4} \frac{sin ( x )}{cos ( x )}

Simplify

(\frac{1}{cos ( x )})^{4} \frac{sin ( x )}{cos ( x )} : \frac{sin ( x )}{cos ^{5} ( x )}

(\frac{1}{cos ( x )})^{4} \frac{sin ( x )}{cos ( x )}

Multiply fractions:

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

= \frac{sin ( x ) ( \frac{1}{c o s ( x )} ) ^{4}}{cos ( x )}

(\frac{1}{cos ( x )})^{4} = \frac{1}{cos ^{4} ( x )}

(\frac{1}{cos ( x )})^{4}

Apply exponent rule:

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

= \frac{1 ^{4}}{cos ^{4} ( x )}

Apply rule

1^{a} = 1

1^{4} = 1

= \frac{1}{cos ^{4} ( x )}

= \frac{\frac{1}{c o s ^{4} ( x )} sin ( x )}{cos ( x )}

Multiply

sin (x) \frac{1}{cos ^{4} ( x )} : \frac{sin ( x )}{cos ^{4} ( x )}

sin (x) \frac{1}{cos ^{4} ( x )}

Multiply fractions:

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

= \frac{1 \cdot sin ( x )}{cos ^{4} ( x )}

Multiply:

1 \cdot sin (x) = sin (x)

= \frac{sin ( x )}{cos ^{4} ( x )}

= \frac{\frac{s i n ( x )}{c o s ^{4} ( x )}}{cos ( x )}

Apply the fraction rule:

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

= \frac{sin ( x )}{cos ^{4} ( x ) cos ( x )}

cos^{4} (x) cos (x) = cos^{5} (x)

cos^{4} (x) cos (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cos^{4} (x) cos (x) = cos^{4 + 1} (x)

= cos^{4 + 1} (x)

Add the numbers:

4 + 1 = 5

= cos^{5} (x)

= \frac{sin ( x )}{cos ^{5} ( x )}

= \frac{sin ( x )}{cos ^{5} ( x )}

= \frac{sin ( x )}{cos ^{5} ( x )}

We showed that the two sides could take the same form

\Rightarrow True

Is tan(x)sec^4(x)=(sin(x))/(cos^5(x)) ?
The answer to whether tan(x)sec^4(x)=(sin(x))/(cos^5(x)) is True