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

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

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

prove

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

True

Solution steps

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

Manipulating right side

2 tan (x) sec (x)

Express with sin, cos

2 sec (x) tan (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

\frac{cos ( θ )}{sin ^{2} ( θ )} = csc (θ) cot (θ)

\frac{1}{2} \frac{tan ( b )}{1 - cos ^{2} ( b )} = csc (2 b)

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

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

\frac{4 + 2 sin ( θ )}{1 - sin ( θ )} = 3

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