What is expand (e^{jt}-e^{-jt})^4 ?

The solution to expand (e^{jt}-e^{-jt})^4 is e^{4jt}-4e^{2jt}+6-4e^{-2jt}+e^{-4jt}

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expand (e^{jt}-e^{-jt})^4

expand

(e^{j t} - e^{- j t})^{4}

e^{4 j t} - 4 e^{2 j t} + 6 - 4 e^{- 2 j t} + e^{- 4 j t}

Solution steps

(e^{j t} - e^{- j t})^{4}

Apply binomial theorem:

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

a = e^{j t}, b = - e^{- j t}

= i = 0 \sum 4 (i 4) (e^{j t})^{(4 - i)} (- e^{- j t})^{i}

Expand summation

= \frac{4 !}{0 ! ( 4 - 0 ) !} (e^{j t})^{4} (- e^{- j t})^{0} + \frac{4 !}{1 ! ( 4 - 1 ) !} (e^{j t})^{3} (- e^{- j t})^{1} + \frac{4 !}{2 ! ( 4 - 2 ) !} (e^{j t})^{2} (- e^{- j t})^{2} + \frac{4 !}{3 ! ( 4 - 3 ) !} (e^{j t})^{1} (- e^{- j t})^{3} + \frac{4 !}{4 ! ( 4 - 4 ) !} (e^{j t})^{0} (- e^{- j t})^{4}

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} (e^{j t})^{4} (- e^{- j t})^{0} : e^{4 j t}

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} (e^{j t})^{3} (- e^{- j t})^{1} : - 4 e^{2 j t}

\frac{4 !}{2 ! ( 4 - 2 ) !} (e^{j t})^{2} (- e^{- j t})^{2} = 6

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} (e^{j t})^{1} (- e^{- j t})^{3} : - 4 e^{- 2 j t}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} (e^{j t})^{0} (- e^{- j t})^{4} : e^{- 4 j t}

= e^{4 j t} - 4 e^{2 j t} + 6 - 4 e^{- 2 j t} + e^{- 4 j t}

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What is expand (e^{jt}-e^{-jt})^4 ?
The solution to expand (e^{jt}-e^{-jt})^4 is e^{4jt}-4e^{2jt}+6-4e^{-2jt}+e^{-4jt}