Is cos(2x)=cos(x+x) ?

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

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prove cos(2x)=cos(x+x)

prove

cos (2 x) = cos (x + x)

True

Solution steps

cos (2 x) = cos (x + x)

Manipulating left side

cos (2 x)

Rewrite using trig identities

cos (2 x)

Use the Double Angle identity:

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

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

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

Manipulating right side

cos (x + x)

Rewrite using trig identities

cos (x + x)

Use the Angle Sum identity:

cos (s + s) = cos (s) cos (s) - sin (s) sin (s)

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

Simplify

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

cos (x) cos (x) - sin (x) sin (x)

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

cos (x) cos (x)

Apply exponent rule:

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

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

= cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cos^{2} (x)

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

sin (x) sin (x)

Apply exponent rule:

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

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

= sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= sin^{2} (x)

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

\frac{csc ( a )}{cot ( a ) + tan ( a )} = cos (a)

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

sec (x) sin (x) = tan (x)

(tan (θ) + cot (θ)) tan (θ) = sec^{2} (θ)

(sin (x)) (sec (x)) = tan (x)