Is csc(w)-sin(w)=cot(w)cos(w) ?

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

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prove csc(w)-sin(w)=cot(w)cos(w)

prove

csc (w) - sin (w) = cot (w) cos (w)

True

Solution steps

csc (w) - sin (w) = cot (w) cos (w)

Manipulating left side

csc (w) - sin (w)

Express with sin, cos

csc (w) - sin (w)

Use the basic trigonometric identity:

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

= \frac{1}{sin ( w )} - sin (w)

Simplify

\frac{1}{sin ( w )} - sin (w) : \frac{1 - sin ^{2} ( w )}{sin ( w )}

\frac{1}{sin ( w )} - sin (w)

Convert element to fraction:

sin (w) = \frac{sin ( w ) sin ( w )}{sin ( w )}

= \frac{1}{sin ( w )} - \frac{sin ( w ) sin ( w )}{sin ( w )}

Since the denominators are equal, combine the fractions:

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

= \frac{1 - sin ( w ) sin ( w )}{sin ( w )}

1 - sin (w) sin (w) = 1 - sin^{2} (w)

1 - sin (w) sin (w)

sin (w) sin (w) = sin^{2} (w)

sin (w) sin (w)

Apply exponent rule:

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

sin (w) sin (w) = sin^{1 + 1} (w)

= sin^{1 + 1} (w)

Add the numbers:

1 + 1 = 2

= sin^{2} (w)

= 1 - sin^{2} (w)

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

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

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

Rewrite using trig identities

\frac{1 - sin ^{2} ( w )}{sin ( w )}

Use the Pythagorean identity:

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

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

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

cos (w) cot (w)

cos (w) cot (w)

= cot (w) cos (w)

We showed that the two sides could take the same form

\Rightarrow True

\frac{csc ( t ) + 1}{cot ( t )} = \frac{cot ( t )}{csc ( t ) - 1}

(3 sin (θ) - 3 cos (θ))^{2} = 9 - 9 sin (2 θ)

\frac{cos ( x )}{sin ^{2} ( x )} = csc (x) cot (x)

tan (2 x) - tan (x) = tan (x) sec (2 x)

tan (x) sin (x) = sec (x) - cos (x)

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