What is expand (1-cot^4(x))/(csc^2(x)(sin(x)+cos(x))) ?

The solution to expand (1-cot^4(x))/(csc^2(x)(sin(x)+cos(x))) is csc(x)(1-cot(x))

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expand (1-cot^4(x))/(csc^2(x)(sin(x)+cos(x)))

expand

\frac{1 - cot ^{4} ( x )}{csc ^{2} ( x ) ( sin ( x ) + cos ( x ) )}

csc (x) (1 - cot (x))

Solution steps

\frac{1 - cot ^{4} ( x )}{csc ^{2} ( x ) ( sin ( x ) + cos ( x ) )}

Expand

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

= \frac{1 - cot ^{4} ( x )}{csc ^{2} ( x ) sin ( x ) + csc ^{2} ( x ) cos ( x )}

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

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

= \frac{1 - cot ^{4} ( x )}{csc ( x ) + cot ( x ) csc ( x )}

Factor

1 - cot^{4} (x) : - (cot^{2} (x) + 1) (cot (x) + 1) (cot (x) - 1)

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

Factor out common term

csc (x)

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

Cancel the common factor:

cot (x) + 1

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

Use the basic trigonometric identity:

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

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

Simplify

cot^{2} (x) + 1 : csc^{2} (x)

= - csc^{2} (x) sin (x) (cot (x) - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = - csc^{2} (x) sin (x), b = cot (x), c = 1

= - csc^{2} (x) sin (x) cot (x) - (- csc^{2} (x) sin (x)) \cdot 1

Apply minus-plus rules

- (- a) = a

= - csc^{2} (x) sin (x) cot (x) + 1 \cdot csc^{2} (x) sin (x)

Multiply:

1 \cdot csc^{2} (x) = csc^{2} (x)

= - csc^{2} (x) sin (x) cot (x) + csc^{2} (x) sin (x)

Use the basic trigonometric identity:

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

= csc^{2} (x) \frac{1}{csc ( x )}

Simplify

csc^{2} (x) \frac{1}{csc ( x )} : csc (x)

= - csc^{2} (x) sin (x) cot (x) + csc (x)

csc (x) sin (x) = 1

= csc (x) (1 - cot (x) \cdot 1)

Factor

- csc^{2} (x) sin (x) cot (x) + csc (x) : csc (x) (- csc (x) sin (x) cot (x) + 1)

= csc (x) (- csc (x) sin (x) cot (x) + 1)

= csc (x) (1 - cot (x))

\frac{x ^{2} + 2 x - 2}{( x - 1 ) ( x ^{2} - 4 x + 2 )}

\frac{e ^{- 2 s}}{( s + 1 ) ( s + 2 )}

x (e^{2 x} + 3 e^{x} = 54)

\frac{( x - 2 ) ( - 4 x - 1 )}{2}

δ (x^{'} - x)

What is expand (1-cot^4(x))/(csc^2(x)(sin(x)+cos(x))) ?
The solution to expand (1-cot^4(x))/(csc^2(x)(sin(x)+cos(x))) is csc(x)(1-cot(x))