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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Use the Double Angle identity:

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

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

Simplify

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

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

Divide the numbers:

\frac{2}{2} = 1

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

Cancel the common factor:

cos (x)

= \frac{cos ( x )}{sin ( x )}

= \frac{cos ( x )}{sin ( x )}

= \frac{cos ( x )}{sin ( x )}

Manipulating right side

\frac{1}{tan ( x )}

Express with sin, cos

\frac{1}{tan ( x )}

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{1}{\frac{s i n ( x )}{c o s ( x )}}

Simplify

\frac{1}{\frac{s i n ( x )}{c o s ( x )}} : \frac{cos ( x )}{sin ( x )}

\frac{1}{\frac{s i n ( x )}{c o s ( x )}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{cos ( x )}{sin ( x )}

= \frac{cos ( x )}{sin ( x )}

= \frac{cos ( x )}{sin ( x )}

We showed that the two sides could take the same form

\Rightarrow True

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

\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)

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