Is 1csc(θ)-(cos(θ))(cot(θ))=sin(θ) ?

The answer to whether 1csc(θ)-(cos(θ))(cot(θ))=sin(θ) is True

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prove 1csc(θ)-(cos(θ))(cot(θ))=sin(θ)

prove

1 csc (θ) - (cos (θ)) (cot (θ)) = sin (θ)

True

Solution steps

1 \cdot csc (θ) - cos (θ) cot (θ) = sin (θ)

Manipulating left side

1 csc (θ) - cos (θ) cot (θ)

Express with sin, cos

1 \cdot csc (θ) - cos (θ) cot (θ)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

1 \cdot \frac{1}{sin ( θ )} - cos (θ) \frac{cos ( θ )}{sin ( θ )}

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

1 \cdot \frac{1}{sin ( θ )}

Multiply:

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

= \frac{1}{sin ( θ )}

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

cos (θ) \frac{cos ( θ )}{sin ( θ )}

Multiply fractions:

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

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

cos (θ) cos (θ) = cos^{2} (θ)

cos (θ) cos (θ)

Apply exponent rule:

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

cos (θ) cos (θ) = cos^{1 + 1} (θ)

= cos^{1 + 1} (θ)

Add the numbers:

1 + 1 = 2

= cos^{2} (θ)

= \frac{cos ^{2} ( θ )}{sin ( θ )}

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

Apply rule

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

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

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

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

Rewrite using trig identities

\frac{1 - cos ^{2} ( θ )}{sin ( θ )}

Use the Pythagorean identity:

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

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

= \frac{sin ^{2} ( θ )}{sin ( θ )}

Cancel the common factor:

sin (θ)

= sin (θ)

= sin (θ)

We showed that the two sides could take the same form

\Rightarrow True

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

sin^{2} (3 x) = 9 sin^{3} (x) cos^{3} (x)

8 tan (x) sin (x) - 8 tan (x) = 0

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

sin (\frac{π}{2} + a) = cos (a)

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