Is (sec(θ)cos(θ))/(tan(θ))=cot(θ) ?

The answer to whether (sec(θ)cos(θ))/(tan(θ))=cot(θ) is True

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prove (sec(θ)cos(θ))/(tan(θ))=cot(θ)

prove

\frac{sec ( θ ) cos ( θ )}{tan ( θ )} = cot (θ)

True

Solution steps

\frac{sec ( θ ) cos ( θ )}{tan ( θ )} = cot (θ)

Manipulating left side

\frac{sec ( θ ) cos ( θ )}{tan ( θ )}

Express with sin, cos

\frac{cos ( θ ) sec ( θ )}{tan ( θ )}

Use the basic trigonometric identity:

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

= \frac{cos ( θ ) \frac{1}{c o s ( θ )}}{tan ( θ )}

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply the fraction rule:

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

= \frac{cos ( θ ) \frac{1}{c o s ( θ )} cos ( θ )}{sin ( θ )}

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

cos (θ) \frac{1}{cos ( θ )} cos (θ)

Apply exponent rule:

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

cos (θ) cos (θ) = cos^{1 + 1} (θ)

= \frac{1}{cos ( θ )} cos^{1 + 1} (θ)

Add the numbers:

1 + 1 = 2

= \frac{1}{cos ( θ )} cos^{2} (θ)

= \frac{\frac{1}{c o s ( θ )} cos ^{2} ( θ )}{sin ( θ )}

Multiply

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

\frac{1}{cos ( θ )} cos^{2} (θ)

Multiply fractions:

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

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

Multiply:

1 \cdot cos^{2} (θ) = cos^{2} (θ)

= \frac{cos ^{2} ( θ )}{cos ( θ )}

Cancel the common factor:

cos (θ)

= cos (θ)

= \frac{cos ( θ )}{sin ( θ )}

= \frac{cos ( θ )}{sin ( θ )}

= \frac{cos ( θ )}{sin ( θ )}

Use the basic trigonometric identity:

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

= cot (θ)

We showed that the two sides could take the same form

\Rightarrow True

tan (\frac{π}{4} + x) = \frac{1 + sin ( 2 x )}{cos ( 2 x )}

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

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

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

sin (x) + cos (x) = 2

Is (sec(θ)cos(θ))/(tan(θ))=cot(θ) ?
The answer to whether (sec(θ)cos(θ))/(tan(θ))=cot(θ) is True