Is 1/(sin(x))=sin(x)+cos(x)cot(x) ?

The answer to whether 1/(sin(x))=sin(x)+cos(x)cot(x) is True

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prove 1/(sin(x))=sin(x)+cos(x)cot(x)

prove

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

True

Solution steps

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

Manipulating right side

sin (x) + cos (x) cot (x)

Express with sin, cos

sin (x) + cos (x) cot (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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 )}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

sin (x) sec (x) = 1

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

tan (a - b) = sin (a - b) sec (a - b)

\frac{csc ( B )}{tan ( B ) + cot ( B )} = cos (B)

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

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