Is csc(x)-cot(x)cos(x)= 1/(csc(x)) ?

The answer to whether csc(x)-cot(x)cos(x)= 1/(csc(x)) is True

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prove csc(x)-cot(x)cos(x)= 1/(csc(x))

prove

csc (x) - cot (x) cos (x) = \frac{1}{csc ( x )}

True

Solution steps

csc (x) - cot (x) cos (x) = \frac{1}{csc ( x )}

Manipulating left side

csc (x) - cot (x) cos (x)

Express with sin, cos

csc (x) - cos (x) cot (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Multiply fractions:

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

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

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

cos (x) cos (x)

Apply exponent rule:

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

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

= cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cos^{2} (x)

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

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

Apply rule

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Cancel the common factor:

sin (x)

= sin (x)

= sin (x)

Use the basic trigonometric identity:

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

csc (A) = \frac{cos ( 2 A )}{sin ( A )} + \frac{sin ( 2 A )}{cos ( A )}

sin^{2} (θ) = 1 + cos^{2} (θ)

sin (\frac{π}{2}) = \frac{4}{2}

sec (\frac{π}{4} + a) sec (\frac{π}{4} - a) = 2 sec (2 a)

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