Is 1/(cos(x))-sin(x)tan(x)=cos(x) ?

The answer to whether 1/(cos(x))-sin(x)tan(x)=cos(x) is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

\frac{1}{cos ( x )} - sin (x) \frac{sin ( x )}{cos ( x )}

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

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

Multiply fractions:

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

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

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

sin (x) sin (x)

Apply exponent rule:

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

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

= sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= sin^{2} (x)

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

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

Apply rule

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Cancel the common factor:

cos (x)

= cos (x)

= cos (x)

We showed that the two sides could take the same form

\Rightarrow True

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

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

sin^{2} (θ) + 2 cos^{2} (θ) - 1 = cos^{2} (θ)

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

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

Is 1/(cos(x))-sin(x)tan(x)=cos(x) ?
The answer to whether 1/(cos(x))-sin(x)tan(x)=cos(x) is True