Is tan^2(x)+tan(x)cot(x)=sec^2(x) ?

The answer to whether tan^2(x)+tan(x)cot(x)=sec^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)+tan(x)cot(x)=sec^2(x)

Solution

prove

tan^{2} (x) + tan (x) cot (x) = sec^{2} (x)

Solution

True

Solution steps

tan^{2} (x) + tan (x) cot (x) = sec^{2} (x)

Manipulating left side

tan^{2} (x) + tan (x) cot (x)

Express with sin, cos

tan^{2} (x) + cot (x) tan (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

\frac{cos ( x )}{sin ( x )} \cdot \frac{sin ( x )}{cos ( x )}

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

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

Cancel the common factor:

sin (x)

= 1

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

Convert element to fraction:

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

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

Multiply:

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Simplify

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= sec^{2} (x)

sec^{2} (x)

sec^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

Popular Examples

prove cos(4θ)=2(cos(2θ))^2-1 prove

cos (4 θ) = 2 (cos (2 θ))^{2} - 1

prove tan(2x)=(2sin(x)cos(x))/(2cos^2(x)-1)prove

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

prove tan(x/2)=-cot(x)+csc(x)prove

tan (\frac{x}{2}) = - cot (x) + csc (x)

prove 1/(csc^2(x))+1/(sec^2(x))=1 prove

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

prove cot((5pi)/(12))=2-sqrt(3)prove

cot (\frac{5 π}{12}) = 2 - 3

Frequently Asked Questions (FAQ)

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