Is (cos(θ)csc(θ))/(cot^2(θ))=tan(θ) ?

The answer to whether (cos(θ)csc(θ))/(cot^2(θ))=tan(θ) is True

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prove (cos(θ)csc(θ))/(cot^2(θ))=tan(θ)

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

= \frac{cos ( θ ) \frac{1}{s i n ( θ )}}{cot ^{2} ( θ )}

Use the basic trigonometric identity:

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

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

Simplify

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

\frac{cos ( θ ) \frac{1}{s i n ( θ )}}{( \frac{c o s ( θ )}{s i n ( θ )} ) ^{2}}

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

(\frac{cos ( θ )}{sin ( θ )})^{2}

Apply exponent rule:

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

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

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

Multiply

cos (θ) \frac{1}{sin ( θ )} : \frac{cos ( θ )}{sin ( θ )}

cos (θ) \frac{1}{sin ( θ )}

Multiply fractions:

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

= \frac{1 \cdot cos ( θ )}{sin ( θ )}

Multiply:

1 \cdot cos (θ) = cos (θ)

= \frac{cos ( θ )}{sin ( θ )}

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

Divide fractions:

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

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

Cancel the common factor:

cos (θ)

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

Cancel the common factor:

sin (θ)

= \frac{sin ( θ )}{cos ( θ )}

= \frac{sin ( θ )}{cos ( θ )}

= \frac{sin ( θ )}{cos ( θ )}

Use the basic trigonometric identity:

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

= tan (θ)

We showed that the two sides could take the same form

\Rightarrow True

cos (\frac{3 π}{2}) = cos (\frac{π}{2})

csc (\frac{π}{2} - x) cos (x) = sec (x) cos (x)

1 + 0 + 1 = - 2 cos (0) + 4

(1 - sin^{2} (y)) (1 + tan^{2} (y)) = 1

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

Is (cos(θ)csc(θ))/(cot^2(θ))=tan(θ) ?
The answer to whether (cos(θ)csc(θ))/(cot^2(θ))=tan(θ) is True