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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

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

Use the Double Angle identity:

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

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

Simplify

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

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

Group like terms

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

1 - 1 = 0

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

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

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

Factor

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

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

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

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

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

Rewrite as

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

Factor out common term

2 sin (x)

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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