Is (2tan(x))/(sec^2(x))=sin(2x) ?

The answer to whether (2tan(x))/(sec^2(x))=sin(2x) is True

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prove (2tan(x))/(sec^2(x))=sin(2x)

prove

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

True

Solution steps

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

Manipulating left side

\frac{2 tan ( x )}{sec ^{2} ( x )}

Express with sin, cos

\frac{2 tan ( x )}{sec ^{2} ( x )}

Use the basic trigonometric identity:

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

= \frac{2 \cdot \frac{s i n ( x )}{c o s ( x )}}{sec ^{2} ( x )}

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= \frac{2 \cdot \frac{s i n ( x )}{c o s ( x )}}{( \frac{1}{c o s ( x )} ) ^{2}}

Simplify

\frac{2 \cdot \frac{s i n ( x )}{c o s ( x )}}{( \frac{1}{c o s ( x )} ) ^{2}} : 2 sin (x) cos (x)

\frac{2 \cdot \frac{s i n ( x )}{c o s ( x )}}{( \frac{1}{c o s ( x )} ) ^{2}}

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

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

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

Apply rule

1^{a} = 1

1^{2} = 1

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

= \frac{2 \cdot \frac{s i n ( x )}{c o s ( x )}}{\frac{1}{c o s ^{2} ( x )}}

Multiply

2 \cdot \frac{sin ( x )}{cos ( x )} : \frac{2 sin ( x )}{cos ( x )}

2 \cdot \frac{sin ( x )}{cos ( x )}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

= \frac{\frac{2 s i n ( x )}{c o s ( x )}}{\frac{1}{c o s ^{2} ( x )}}

Divide fractions:

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}

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

Refine

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

Cancel the common factor:

cos (x)

= 2 sin (x) cos (x)

= 2 sin (x) cos (x)

= 2 cos (x) sin (x)

Rewrite using trig identities

2 cos (x) sin (x)

Use the Double Angle identity:

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

= sin (2 x)

= sin (2 x)

We showed that the two sides could take the same form

\Rightarrow True

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

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

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

2 csc (2 θ) cos (2 θ) = cot (θ) - tan (θ)

6 sin (x) cos (x) = 3 sin (2 x)

Is (2tan(x))/(sec^2(x))=sin(2x) ?
The answer to whether (2tan(x))/(sec^2(x))=sin(2x) is True