Is (csc(θ)sin(θ))/(cot(θ))=tan(θ) ?

The answer to whether (csc(θ)sin(θ))/(cot(θ))=tan(θ) is True

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prove (csc(θ)sin(θ))/(cot(θ))=tan(θ)

prove

\frac{csc ( θ ) sin ( θ )}{cot ( θ )} = tan (θ)

True

Solution steps

\frac{csc ( θ ) sin ( θ )}{cot ( θ )} = tan (θ)

Manipulating left side

\frac{csc ( θ ) sin ( θ )}{cot ( θ )}

Express with sin, cos

\frac{csc ( θ ) sin ( θ )}{cot ( θ )}

Use the basic trigonometric identity:

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

= \frac{\frac{1}{s i n ( θ )} sin ( θ )}{cot ( θ )}

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply the fraction rule:

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

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

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

\frac{1}{sin ( θ )} sin (θ) sin (θ)

Apply exponent rule:

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

sin (θ) sin (θ) = sin^{1 + 1} (θ)

= \frac{1}{sin ( θ )} sin^{1 + 1} (θ)

Add the numbers:

1 + 1 = 2

= \frac{1}{sin ( θ )} sin^{2} (θ)

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

Multiply

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

\frac{1}{sin ( θ )} sin^{2} (θ)

Multiply fractions:

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

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

Multiply:

1 \cdot sin^{2} (θ) = sin^{2} (θ)

= \frac{sin ^{2} ( θ )}{sin ( θ )}

Cancel the common factor:

sin (θ)

= 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

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

\frac{sin ( 9 θ ) + sin ( 3 θ )}{cos ( 9 θ ) + cos ( 3 θ )} = tan (6 θ)

cot^{2} (A) - cos^{2} (A) = cot^{2} (A) cos^{2} (A)

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

tan (α) + cot (2 α) = sec (2 α)

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