Is 1/(cos(θ))-cos(θ)=sin(θ)tan(θ) ?

The answer to whether 1/(cos(θ))-cos(θ)=sin(θ)tan(θ) is True

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prove 1/(cos(θ))-cos(θ)=sin(θ)tan(θ)

prove

\frac{1}{cos ( θ )} - cos (θ) = sin (θ) tan (θ)

True

Solution steps

\frac{1}{cos ( θ )} - cos (θ) = sin (θ) tan (θ)

Manipulating right side

sin (θ) tan (θ)

Express with sin, cos

sin (θ) tan (θ)

Use the basic trigonometric identity:

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

= sin (θ) \frac{sin ( θ )}{cos ( θ )}

Simplify

sin (θ) \frac{sin ( θ )}{cos ( θ )} : \frac{sin ^{2} ( θ )}{cos ( θ )}

sin (θ) \frac{sin ( θ )}{cos ( θ )}

Multiply fractions:

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

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

sin (θ) sin (θ) = sin^{2} (θ)

sin (θ) sin (θ)

Apply exponent rule:

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

sin (θ) sin (θ) = sin^{1 + 1} (θ)

= sin^{1 + 1} (θ)

Add the numbers:

1 + 1 = 2

= sin^{2} (θ)

= \frac{sin ^{2} ( θ )}{cos ( θ )}

= \frac{sin ^{2} ( θ )}{cos ( θ )}

= \frac{sin ^{2} ( θ )}{cos ( θ )}

Rewrite using trig identities

\frac{sin ^{2} ( θ )}{cos ( θ )}

Use the Pythagorean identity:

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

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

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

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

Expand

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

\frac{1 - cos ^{2} ( θ )}{cos ( θ )}

Apply the fraction rule:

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

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

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

Cancel

\frac{cos ^{2} ( θ )}{cos ( θ )} : cos (θ)

\frac{cos ^{2} ( θ )}{cos ( θ )}

Cancel the common factor:

cos (θ)

= cos (θ)

= \frac{1}{cos ( θ )} - cos (θ)

= - cos (θ) + \frac{1}{cos ( θ )}

= \frac{1}{cos ( θ )} - cos (θ)

We showed that the two sides could take the same form

\Rightarrow True

Is 1/(cos(θ))-cos(θ)=sin(θ)tan(θ) ?
The answer to whether 1/(cos(θ))-cos(θ)=sin(θ)tan(θ) is True