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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

2 - 2 sin (x) - (1 - sin^{2} (x)) : sin^{2} (x) - 2 sin (x) + 1

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

- (1 - sin^{2} (x)) : - 1 + sin^{2} (x)

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

Distribute parentheses

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

Apply minus-plus rules

- (- a) = a, - (a) = - a

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

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

Simplify

2 - 2 sin (x) - 1 + sin^{2} (x) : sin^{2} (x) - 2 sin (x) + 1

2 - 2 sin (x) - 1 + sin^{2} (x)

Group like terms

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

Add/Subtract the numbers:

2 - 1 = 1

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

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

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

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

Manipulating right side

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

Expand

(- 1 + sin (x))^{2} : 1 - 2 sin (x) + sin^{2} (x)

(- 1 + sin (x))^{2}

Apply Perfect Square Formula:

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

a = - 1, b = sin (x)

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

Simplify

(- 1)^{2} + 2 (- 1) sin (x) + sin^{2} (x) : 1 - 2 sin (x) + sin^{2} (x)

(- 1)^{2} + 2 (- 1) sin (x) + sin^{2} (x)

Remove parentheses:

(- a) = - a

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

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 1)^{2} = 1^{2}

= 1^{2}

Apply rule

1^{a} = 1

= 1

2 \cdot 1 \cdot sin (x) = 2 sin (x)

2 \cdot 1 \cdot sin (x)

Multiply the numbers:

2 \cdot 1 = 2

= 2 sin (x)

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

cos (4 x) = 1 - 8 sin^{2} (x) cos (x)

\frac{( 1 )}{( 2 )} (cot (θ) - tan (θ)) = cot (2 θ)

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

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

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

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