Is 3-4cos^2(x)=(2sin(x)+1)(2sin(x)-1) ?

The answer to whether 3-4cos^2(x)=(2sin(x)+1)(2sin(x)-1) is True

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prove 3-4cos^2(x)=(2sin(x)+1)(2sin(x)-1)

prove

3 - 4 cos^{2} (x) = (2 sin (x) + 1) (2 sin (x) - 1)

True

Solution steps

3 - 4 cos^{2} (x) = (2 sin (x) + 1) (2 sin (x) - 1)

Manipulating left side

3 - 4 cos^{2} (x)

Rewrite using trig identities

3 - 4 cos^{2} (x)

Use the Pythagorean identity:

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

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

= 3 - 4 (1 - sin^{2} (x))

Simplify

3 - 4 (1 - sin^{2} (x)) : 4 sin^{2} (x) - 1

3 - 4 (1 - sin^{2} (x))

Expand

- 4 (1 - sin^{2} (x)) : - 4 + 4 sin^{2} (x)

- 4 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 4, b = 1, c = sin^{2} (x)

= - 4 \cdot 1 - (- 4) sin^{2} (x)

Apply minus-plus rules

- (- a) = a

= - 4 \cdot 1 + 4 sin^{2} (x)

Multiply the numbers:

4 \cdot 1 = 4

= - 4 + 4 sin^{2} (x)

= 3 - 4 + 4 sin^{2} (x)

Subtract the numbers:

3 - 4 = - 1

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

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

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

Manipulating right side

(2 sin (x) + 1) (2 sin (x) - 1)

Expand

(- 1 + 2 sin (x)) (1 + 2 sin (x)) : 4 sin^{2} (x) - 1

(- 1 + 2 sin (x)) (1 + 2 sin (x))

Apply Difference of Two Squares Formula:

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

a = 2 sin (x), b = 1

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

Simplify

(2 sin (x))^{2} - 1^{2} : 4 sin^{2} (x) - 1

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

Apply rule

1^{a} = 1

1^{2} = 1

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

(2 sin (x))^{2} = 4 sin^{2} (x)

(2 sin (x))^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

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

2^{2} = 4

= 4 sin^{2} (x)

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

\frac{1 + tan ( x )}{1 + \frac{1}{t a n ( x )}} = tan (x)

cos (2 θ) = \frac{1}{sec ( 2 θ )}

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

(cos (3 x) - cos (x)) = - 2 sin (2 x) sin (x)

Is 3-4cos^2(x)=(2sin(x)+1)(2sin(x)-1) ?
The answer to whether 3-4cos^2(x)=(2sin(x)+1)(2sin(x)-1) is True