Is tan^2(x)=(sec^2(x))/(csc^2(x)) ?

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

prove

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

True

Solution steps

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

Manipulating left side

tan^{2} (x)

Express with sin, cos

tan^{2} (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

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

Divide fractions:

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

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

Refine

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

tan (x) cot (x) sin (x) = csc (x)

\frac{7}{tan ( x )} + \frac{7}{cot ( x )} = 7 tan (x) + 7 cot (x)

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

4 tan (x) + 2 sin (x) cos (x) = 0

2 cos^{2} (x) - 3 cos (x) + 1 = 0

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