Is sec(u)cos(u)-sin^2(u)=cos^2(u) ?

The answer to whether sec(u)cos(u)-sin^2(u)=cos^2(u) is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

prove sec(u)cos(u)-sin^2(u)=cos^2(u)

prove

sec (u) cos (u) - sin^{2} (u) = cos^{2} (u)

True

Solution steps

sec (u) cos (u) - sin^{2} (u) = cos^{2} (u)

Manipulating left side

sec (u) cos (u) - sin^{2} (u)

Express with sin, cos

- sin^{2} (u) + cos (u) sec (u)

Use the basic trigonometric identity:

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

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

Simplify

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

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

cos (u) \frac{1}{cos ( u )} = 1

cos (u) \frac{1}{cos ( u )}

Multiply fractions:

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

= \frac{1 \cdot cos ( u )}{cos ( u )}

Cancel the common factor:

cos (u)

= 1

= - sin^{2} (u) + 1

= - sin^{2} (u) + 1

= 1 - sin^{2} (u)

Rewrite using trig identities

1 - sin^{2} (u)

Use the Pythagorean identity:

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

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

= cos^{2} (u)

= cos^{2} (u)

We showed that the two sides could take the same form

\Rightarrow True

Is sec(u)cos(u)-sin^2(u)=cos^2(u) ?
The answer to whether sec(u)cos(u)-sin^2(u)=cos^2(u) is True