Is sin(θ)+(cos(θ))/(tan(θ))=csc(θ) ?

The answer to whether sin(θ)+(cos(θ))/(tan(θ))=csc(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)+(cos(θ))/(tan(θ))=csc(θ)

prove

sin (θ) + \frac{cos ( θ )}{tan ( θ )} = csc (θ)

True

Solution steps

sin (θ) + \frac{cos ( θ )}{tan ( θ )} = csc (θ)

Manipulating left side

sin (θ) + \frac{cos ( θ )}{tan ( θ )}

Express with sin, cos

\frac{cos ( θ )}{tan ( θ )} + sin (θ)

Use the basic trigonometric identity:

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

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

Simplify

\frac{cos ( θ )}{\frac{s i n ( θ )}{c o s ( θ )}} + sin (θ) : \frac{cos ^{2} ( θ ) + sin ^{2} ( θ )}{sin ( θ )}

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

\frac{cos ( θ )}{\frac{s i n ( θ )}{c o s ( θ )}} = \frac{cos ^{2} ( θ )}{sin ( θ )}

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

cos (θ) cos (θ) = cos^{2} (θ)

cos (θ) cos (θ)

Apply exponent rule:

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

cos (θ) cos (θ) = cos^{1 + 1} (θ)

= cos^{1 + 1} (θ)

Add the numbers:

1 + 1 = 2

= cos^{2} (θ)

= \frac{cos ^{2} ( θ )}{sin ( θ )}

= \frac{cos ^{2} ( θ )}{sin ( θ )} + sin (θ)

Convert element to fraction:

sin (θ) = \frac{sin ( θ ) sin ( θ )}{sin ( θ )}

= \frac{cos ^{2} ( θ )}{sin ( θ )} + \frac{sin ( θ ) sin ( θ )}{sin ( θ )}

Since the denominators are equal, combine the fractions:

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

= \frac{cos ^{2} ( θ ) + sin ( θ ) sin ( θ )}{sin ( θ )}

cos^{2} (θ) + sin (θ) sin (θ) = cos^{2} (θ) + sin^{2} (θ)

cos^{2} (θ) + sin (θ) sin (θ)

sin (θ) sin (θ) = sin^{2} (θ)

sin (θ) sin (θ)

Apply exponent rule:

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

sin (θ) sin (θ) = sin^{1 + 1} (θ)

= sin^{1 + 1} (θ)

Add the numbers:

1 + 1 = 2

= sin^{2} (θ)

= cos^{2} (θ) + sin^{2} (θ)

= \frac{cos ^{2} ( θ ) + sin ^{2} ( θ )}{sin ( θ )}

= \frac{cos ^{2} ( θ ) + sin ^{2} ( θ )}{sin ( θ )}

= \frac{cos ^{2} ( θ ) + sin ^{2} ( θ )}{sin ( θ )}

Rewrite using trig identities

\frac{cos ^{2} ( θ ) + sin ^{2} ( θ )}{sin ( θ )}

Use the Pythagorean identity:

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

= \frac{1}{sin ( θ )}

= \frac{1}{sin ( θ )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{\frac{1}{c s c ( θ )}}

Simplify

\frac{1}{\frac{1}{c s c ( θ )}}

Apply the fraction rule:

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

= \frac{csc ( θ )}{1}

Apply rule

\frac{a}{1} = a

= csc (θ)

csc (θ)

csc (θ)

We showed that the two sides could take the same form

\Rightarrow True

(1 + tan^{2} (u)) (1 - sin^{2} (u)) = 1

sec^{3} (x) = \frac{1}{cos ^{3} ( x )}

sin (x) = \frac{5}{13}

sec (x) - tan (x) \cdot sin (x) = \frac{1}{sec ( x )}

sin^{4} (x) + cos^{4} (x) = 1 - \frac{1}{2} sin^{2} (2 x)

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