Is sin(x)-csc(x)=(-cos^2(x))/(sin(x)) ?

The answer to whether sin(x)-csc(x)=(-cos^2(x))/(sin(x)) is True

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prove sin(x)-csc(x)=(-cos^2(x))/(sin(x))

prove

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

True

Solution steps

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

Manipulating left side

sin (x) - csc (x)

Express with sin, cos

- csc (x) + sin (x)

Use the basic trigonometric identity:

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

= - \frac{1}{sin ( x )} + sin (x)

Simplify

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

- \frac{1}{sin ( x )} + sin (x)

Convert element to fraction:

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

= - \frac{1}{sin ( x )} + \frac{sin ( x ) sin ( x )}{sin ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{- 1 + sin ( x ) sin ( x )}{sin ( x )}

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

- 1 + sin (x) sin (x)

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

sin (x) sin (x)

Apply exponent rule:

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

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

= sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= sin^{2} (x)

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

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

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

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

Rewrite using trig identities

\frac{- 1 + sin ^{2} ( x )}{sin ( x )}

Use the Pythagorean identity:

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

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

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

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

sec^{2} (x) - sec^{2} (x) cos^{2} (x) = tan^{2} (x)

\frac{1}{1 - cos ^{2} ( x )} = 1 + cot^{2} (x)

\frac{2}{tan ( x ) + cot ( x )} = sin (2 x)

sin (x) tan (x) sec (x) = sec^{2} (x) - 1

tan^{4} (k) - sec^{4} (k) = 1 - 2 sec^{2} (k)

Is sin(x)-csc(x)=(-cos^2(x))/(sin(x)) ?
The answer to whether sin(x)-csc(x)=(-cos^2(x))/(sin(x)) is True