What is expand (1-tan(x))^5 ?

The solution to expand (1-tan(x))^5 is 1-5tan(x)+10tan^2(x)-10tan^3(x)+5tan^4(x)-tan^5(x)

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expand (1-tan(x))^5

expand

(1 - tan (x))^{5}

1 - 5 tan (x) + 10 tan^{2} (x) - 10 tan^{3} (x) + 5 tan^{4} (x) - tan^{5} (x)

Solution steps

(1 - tan (x))^{5}

Apply binomial theorem:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = 1, b = - tan (x)

= i = 0 \sum 5 (i 5) \cdot 1^{(5 - i)} (- tan (x))^{i}

Expand summation

= \frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 1^{5} (- tan (x))^{0} + \frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 1^{4} (- tan (x))^{1} + \frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 1^{3} (- tan (x))^{2} + \frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 1^{2} (- tan (x))^{3} + \frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 1^{1} (- tan (x))^{4} + \frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 1^{0} (- tan (x))^{5}

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 1^{5} (- tan (x))^{0} : 1

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 1^{4} (- tan (x))^{1} : - 5 tan (x)

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 1^{3} (- tan (x))^{2} : 10 tan^{2} (x)

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 1^{2} (- tan (x))^{3} : - 10 tan^{3} (x)

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 1^{1} (- tan (x))^{4} : 5 tan^{4} (x)

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 1^{0} (- tan (x))^{5} : - tan^{5} (x)

= 1 - 5 tan (x) + 10 tan^{2} (x) - 10 tan^{3} (x) + 5 tan^{4} (x) - tan^{5} (x)

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(sin (x) - \frac{sin ^{3} ( x )}{3}) sin (x)

What is expand (1-tan(x))^5 ?
The solution to expand (1-tan(x))^5 is 1-5tan(x)+10tan^2(x)-10tan^3(x)+5tan^4(x)-tan^5(x)