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

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

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

prove

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

True

Solution steps

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

Manipulating left side

2 sin^{2} (2 x) + cos (4 x)

Rewrite using trig identities

2 sin^{2} (2 x) + cos (4 x)

Use the Double Angle identity:

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

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

= cos (4 x) - (cos (2 \cdot 2 x) - 1)

cos (4 x) - (cos (2 \cdot 2 x) - 1) = 1

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

Multiply the numbers:

2 \cdot 2 = 4

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

- (cos (4 x) - 1) : - cos (4 x) + 1

- (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) - cos (4 x) + 1

Add similar elements:

cos (4 x) - cos (4 x) = 0

= 1

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

\frac{sin ( x )}{cos ( x ) cot ^{2} ( x )} = tan (x)

cos (θ) - cos (θ) sin^{2} (θ) = cos^{3} (θ)

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

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

cot (x) tan (x) = (sec (x)) (csc (x))