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

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

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

prove

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

True

Solution steps

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

Manipulating left side

(cos (x) + 1) (cos (x) - 1)

Expand

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

(cos (x) + 1) (cos (x) - 1)

Apply Difference of Two Squares Formula:

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

a = cos (x), b = 1

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

Apply rule

1^{a} = 1

1^{2} = 1

= cos^{2} (x) - 1

= cos^{2} (x) - 1

Use the Pythagorean identity:

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

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

= - sin^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

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

sin (θ - \frac{π}{3}) + cos (θ + \frac{11 π}{6}) = sin (θ)

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

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

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

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