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

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

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

prove

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

True

Solution steps

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

Manipulating right side

2 sin^{2} (2 x)

Rewrite using trig identities

2 sin^{2} (2 x)

Use the Double Angle identity:

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

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

= - (cos (2 \cdot 2 x) - 1)

Simplify

- (cos (2 \cdot 2 x) - 1) : - cos (4 x) + 1

- (cos (2 \cdot 2 x) - 1)

Multiply the numbers:

2 \cdot 2 = 4

= - (cos (4 x) - 1)

Distribute parentheses

= - (cos (4 x)) - (- 1)

Apply minus-plus rules

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

= - cos (4 x) + 1

= - cos (4 x) + 1

= - cos (4 x) + 1

= 1 - cos (4 x)

We showed that the two sides could take the same form

\Rightarrow True

(1 - cos (x)) (1 + sec (x)) = sin (x) tan (x)

cos (2 β) = \frac{1 - tan ^{2} ( β )}{1 + tan ^{2} ( β )}

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

sec (x) - tan (x) = \frac{1}{sec ( x )}

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