Is csc^2(x)=(tan^2(x)+1)/(tan^2(x)) ?

The answer to whether csc^2(x)=(tan^2(x)+1)/(tan^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^2(x)=(tan^2(x)+1)/(tan^2(x))

Solution

prove

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

Solution

True

Solution steps

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

Manipulating right side

\frac{tan ^{2} ( x ) + 1}{tan ^{2} ( x )}

Express with sin, cos

\frac{1 + tan ^{2} ( x )}{tan ^{2} ( x )}

Use the basic trigonometric identity:

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

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

Simplify

\frac{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{( \frac{s i n ( x )}{c o s ( x )} ) ^{2}} : \frac{cos ^{2} ( x ) + sin ^{2} ( x )}{sin ^{2} ( x )}

\frac{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}

(\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{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{\frac{s i n ^{2} ( x )}{c o s ^{2} ( x )}}

Apply exponent rule:

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

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

Apply the fraction rule:

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

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

Join

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

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

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

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

Multiply

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos^{2} (x)

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{( \frac{1}{c s c ( x )} ) ^{2}}

Simplify

\frac{1}{( \frac{1}{c s c ( x )} ) ^{2}}

(\frac{1}{csc ( x )})^{2} = \frac{1}{csc ^{2} ( x )}

(\frac{1}{csc ( x )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{csc ^{2} ( x )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{csc ^{2} ( x )}

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

Apply the fraction rule:

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

= \frac{csc ^{2} ( x )}{1}

Apply rule

\frac{a}{1} = a

= csc^{2} (x)

csc^{2} (x)

csc^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

Popular Examples

prove 2cot(2θ)=cot(θ)-tan(θ)prove

2 cot (2 θ) = cot (θ) - tan (θ)

prove 7cos^2(b)-7sin^2(b)=14cos^2(b)-7 prove

7 cos^{2} (b) - 7 sin^{2} (b) = 14 cos^{2} (b) - 7

prove 1+tan(x)*tan(2x)=sec(2x)prove

1 + tan (x) \cdot tan (2 x) = sec (2 x)

prove 0sec(0)=tan(0)prove

0 sec (0) = tan (0)

prove tan^3(x)(csc^3(x))=sec^3(x)prove

tan^{3} (x) (csc^{3} (x)) = sec^{3} (x)

Frequently Asked Questions (FAQ)

Is csc^2(x)=(tan^2(x)+1)/(tan^2(x)) ?
The answer to whether csc^2(x)=(tan^2(x)+1)/(tan^2(x)) is True