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

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

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

prove

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

True

Solution steps

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

Manipulating left side

csc (s) - sin (s)

Express with sin, cos

csc (s) - sin (s)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

1 - sin (s) sin (s)

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

sin (s) sin (s)

Apply exponent rule:

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

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

= sin^{1 + 1} (s)

Add the numbers:

1 + 1 = 2

= sin^{2} (s)

= 1 - sin^{2} (s)

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

cos (s) cot (s)

cos (s) cot (s)

We showed that the two sides could take the same form

\Rightarrow True

p ro v e \frac{( 4 sin ^{2} ( x ) )}{tan ^{2} ( x )} = 4 cos^{2} (x)

p ro v e \frac{tan ( 2 x ) + tan ( x )}{1 - tan ( 2 x ) tan ( x )} = tan (3 x)

p ro v e csc (x) \cdot (sin (x) + tan (x)) = 1 + sec (x)

p ro v e 1 - \frac{1 - cos ^{2} ( x )}{1 - cos ( x )} = - cos (x)

p ro v e \frac{sin ( 4 θ ) + sin ( 2 θ )}{cos ( 4 θ ) + cos ( 2 θ )} = tan (3 θ)

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