Is csc(x)tan^2(x)=(sin(x))/(cos^2(x)) ?

The answer to whether csc(x)tan^2(x)=(sin(x))/(cos^2(x)) is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove csc(x)tan^2(x)=(sin(x))/(cos^2(x))

prove

csc (x) tan^{2} (x) = \frac{sin ( x )}{cos ^{2} ( x )}

True

Solution steps

csc (x) tan^{2} (x) = \frac{sin ( x )}{cos ^{2} ( x )}

Manipulating left side

csc (x) tan^{2} (x)

Express with sin, cos

csc (x) tan^{2} (x)

Use the basic trigonometric identity:

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

= \frac{1}{sin ( x )} tan^{2} (x)

Use the basic trigonometric identity:

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

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

Simplify

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

\frac{1}{sin ( x )} (\frac{sin ( x )}{cos ( x )})^{2}

Multiply fractions:

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

= \frac{1 \cdot ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{sin ( x )}

Multiply:

1 \cdot (\frac{sin ( x )}{cos ( x )})^{2} = (\frac{sin ( x )}{cos ( x )})^{2}

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

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

(\frac{sin ( x )}{cos ( x )})^{2}

Apply exponent rule:

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

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

sin (x)

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

\frac{tan ( x )}{sin ( x )} = \frac{1}{cos ( x )}

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

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

(csc (x) - cot (x)) (sec (x) 1) = tan (x)

- \frac{cos ^{2} ( x )}{sin ( x )} = - cos (x) cot (x)

Is csc(x)tan^2(x)=(sin(x))/(cos^2(x)) ?
The answer to whether csc(x)tan^2(x)=(sin(x))/(cos^2(x)) is True