Is (sec^2(t))/(csc(t))=sec(t)tan(t) ?

The answer to whether (sec^2(t))/(csc(t))=sec(t)tan(t) is True

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prove (sec^2(t))/(csc(t))=sec(t)tan(t)

Solution

prove

\frac{sec ^{2} ( t )}{csc ( t )} = sec (t) tan (t)

Solution

True

Solution steps

\frac{sec ^{2} ( t )}{csc ( t )} = sec (t) tan (t)

Manipulating right side

sec (t) tan (t)

Express with sin, cos

sec (t) tan (t)

Use the basic trigonometric identity:

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

= \frac{1}{cos ( t )} tan (t)

Use the basic trigonometric identity:

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

= \frac{1}{cos ( t )} \cdot \frac{sin ( t )}{cos ( t )}

Simplify

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

\frac{1}{cos ( t )} \cdot \frac{sin ( t )}{cos ( t )}

Multiply fractions:

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

= \frac{1 \cdot sin ( t )}{cos ( t ) cos ( t )}

Multiply:

1 \cdot sin (t) = sin (t)

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

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

cos (t) cos (t)

Apply exponent rule:

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

cos (t) cos (t) = cos^{1 + 1} (t)

= cos^{1 + 1} (t)

Add the numbers:

1 + 1 = 2

= cos^{2} (t)

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

sin (x) = \frac{1}{csc ( x )}

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

Use the basic trigonometric identity:

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

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

Simplify

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

Apply the fraction rule:

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

= \frac{1}{csc ( t ) ( \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 )} csc ( t )}

Multiply

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

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

Multiply fractions:

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

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

Multiply:

1 \cdot csc (t) = csc (t)

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

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

Apply the fraction rule:

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is (sec^2(t))/(csc(t))=sec(t)tan(t) ?
The answer to whether (sec^2(t))/(csc(t))=sec(t)tan(t) is True