Is (1+cos(2x))/(cos(x))=2cos(x) ?

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

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prove (1+cos(2x))/(cos(x))=2cos(x)

prove

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

True

Solution steps

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

Manipulating left side

\frac{1 + cos ( 2 x )}{cos ( x )}

Rewrite using trig identities

\frac{1 + cos ( 2 x )}{cos ( x )}

Use the Double Angle identity:

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

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

Simplify

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

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

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

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

Group like terms

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

1 - 1 = 0

= 2 cos^{2} (x)

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

Cancel the common factor:

cos (x)

= 2 cos (x)

= 2 cos (x)

= 2 cos (x)

We showed that the two sides could take the same form

\Rightarrow True

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