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Popular >

prove (sin(t))/(csc(t))=1-(cos(t))/(sec(t))

Solution

prove

\frac{sin ( t )}{csc ( t )} = 1 - \frac{cos ( t )}{sec ( t )}

Solution

True

Solution steps

\frac{sin ( t )}{csc ( t )} = 1 - \frac{cos ( t )}{sec ( t )}

Manipulating left side

\frac{sin ( t )}{csc ( t )}

Express with sin, cos

\frac{sin ( t )}{csc ( t )}

Use the basic trigonometric identity:

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

= \frac{sin ( t )}{\frac{1}{s i n ( t )}}

Simplify

\frac{sin ( t )}{\frac{1}{s i n ( t )}} : sin^{2} (t)

\frac{sin ( t )}{\frac{1}{s i n ( t )}}

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= sin (t) sin (t)

Apply exponent rule:

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

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

= sin^{1 + 1} (t)

Add the numbers:

1 + 1 = 2

= sin^{2} (t)

= sin^{2} (t)

= sin^{2} (t)

Manipulating right side

1 - \frac{cos ( t )}{sec ( t )}

Express with sin, cos

1 - \frac{cos ( t )}{sec ( t )}

Use the basic trigonometric identity:

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

= 1 - \frac{cos ( t )}{\frac{1}{c o s ( t )}}

Simplify

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

1 - \frac{cos ( t )}{\frac{1}{c o s ( t )}}

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

\frac{cos ( t )}{\frac{1}{c o s ( t )}}

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= 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)

= 1 - cos^{2} (t)

= 1 - cos^{2} (t)

= 1 - cos^{2} (t)

Rewrite using trig identities

1 - cos^{2} (t)

Use the Pythagorean identity:

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

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

= sin^{2} (t)

= sin^{2} (t)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is (sin(t))/(csc(t))=1-(cos(t))/(sec(t)) ?
The answer to whether (sin(t))/(csc(t))=1-(cos(t))/(sec(t)) is True