Is 1-2sin(θ)cos(θ)=(cos(θ)-sin(θ))^2 ?

The answer to whether 1-2sin(θ)cos(θ)=(cos(θ)-sin(θ))^2 is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove 1-2sin(θ)cos(θ)=(cos(θ)-sin(θ))^2

prove

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

True

Solution steps

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

Manipulating right side

(cos (θ) - sin (θ))^{2}

Expand

(cos (θ) - sin (θ))^{2} : cos^{2} (θ) - 2 cos (θ) sin (θ) + sin^{2} (θ)

(cos (θ) - sin (θ))^{2}

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = cos (θ), b = sin (θ)

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

= 1 - 2 cos (θ) sin (θ)

= 1 - 2 cos (θ) sin (θ)

= 1 - 2 sin (θ) cos (θ)

We showed that the two sides could take the same form

\Rightarrow True

9 cos^{2} (x) - 9 sin^{2} (x) = 9 - 18 sin^{2} (x)

cos^{2} (x) tan^{2} (x) csc (x) = sin (x)

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

sec^{2} (x) (sec (x) - cos (x)) = tan^{2} (x)

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

Is 1-2sin(θ)cos(θ)=(cos(θ)-sin(θ))^2 ?
The answer to whether 1-2sin(θ)cos(θ)=(cos(θ)-sin(θ))^2 is True