Is (sin(θ)csc(θ))/(tan(θ))=cot(θ) ?

The answer to whether (sin(θ)csc(θ))/(tan(θ))=cot(θ) is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove (sin(θ)csc(θ))/(tan(θ))=cot(θ)

prove

\frac{sin ( θ ) csc ( θ )}{tan ( θ )} = cot (θ)

True

Solution steps

\frac{sin ( θ ) csc ( θ )}{tan ( θ )} = cot (θ)

Manipulating left side

\frac{sin ( θ ) csc ( θ )}{tan ( θ )}

Express with sin, cos

\frac{csc ( θ ) sin ( θ )}{tan ( θ )}

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

= \frac{\frac{1}{s i n ( θ )} sin ( θ )}{\frac{s i n ( θ )}{c o s ( θ )}}

Simplify

\frac{\frac{1}{s i n ( θ )} sin ( θ )}{\frac{s i n ( θ )}{c o s ( θ )}} : \frac{cos ( θ )}{sin ( θ )}

\frac{\frac{1}{s i n ( θ )} sin ( θ )}{\frac{s i n ( θ )}{c o s ( θ )}}

Apply the fraction rule:

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

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

Multiply

\frac{1}{sin ( θ )} sin (θ) cos (θ) : cos (θ)

\frac{1}{sin ( θ )} sin (θ) cos (θ)

Multiply fractions:

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

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

Cancel the common factor:

sin (θ)

= 1 \cdot cos (θ)

Multiply:

1 \cdot cos (θ) = cos (θ)

= cos (θ)

= \frac{cos ( θ )}{sin ( θ )}

= \frac{cos ( θ )}{sin ( θ )}

= \frac{cos ( θ )}{sin ( θ )}

Use the basic trigonometric identity:

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

= cot (θ)

We showed that the two sides could take the same form

\Rightarrow True

sec^{2} (θ) + csc (θ) = - sec^{2} (θ) csc (θ)

tan (5 x) = tan (3 x) sec (2 x) - tan (3 x)

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

sin (4 0^{\circ}) = 2 \cdot sin (2 0^{\circ}) \cdot cos (2 0^{\circ})

9 sec (y) cos (y) = 9

Is (sin(θ)csc(θ))/(tan(θ))=cot(θ) ?
The answer to whether (sin(θ)csc(θ))/(tan(θ))=cot(θ) is True