Prove Trig Identity: $\sin^7 x=\dfrac{35\sin x-21\sin 3x+7\sin 5x-\sin 7x}{64}$

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• anemone
In summary, proving a trigonometric identity involves demonstrating the equality of two expressions for all values of the variables involved. This process is important in mathematics and science as it validates equations and simplifies complex problems. The process includes using algebraic manipulations and trigonometric identities such as Pythagorean, double angle, and sum and difference identities. It is crucial to show all steps in order to ensure transparency and thoroughness, and to avoid common mistakes such as using incorrect identities or forgetting to simplify both sides of the equation. Helpful tips include using Pythagorean identities, double angle identities, and substitutions, while being cautious with odd powers of trigonometric functions.
anemone
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MHB
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Prove that $\sin^7 x=\dfrac{35\sin x-21\sin 3x+7\sin 5x-\sin 7x}{64}$.

We have $\sin\,x=\frac{e^{ix}-e^{-ix}}{2i}$

To avoid fraction we have

$2i\sin\,x=e^{ix}-e^{-ix}$

Take power 7

$-128i\sin^7x=(e^{ix}-e^{-ix})^7$

$={7\choose 0}e^{7ix}- {7\choose 1}e^{5ix} + {7\choose 2}e^{3ix} - {7\choose 3}e^{ix} + {7\choose 4}e^{-ix} - {7\choose 5}e^{-3ix} + {7\choose 6}e^{-5ix}- {7\choose 7}e^{-7ix}$

$={7\choose 0}e^{7ix}- {7\choose 1}e^{5ix} + {7\choose 2}e^{3ix} - {7\choose 3}e^{ix} + {7\choose 3}e^{-ix} - {7\choose 2}e^{-3ix} + {7\choose 1}e^{-5ix} - {7\choose 0}e^{-7ix}$ using ${n\choose r} = {n\choose (n-r)}$

$={7\choose 0}(e^{7ix}-e^{-7ix}) - {7\choose 1}(e^{5ix} -e^{5ix}) + {7\choose 2}(e^{3ix} - e^{-3ix}) - {7\choose 3}(e^{ix} - e^{-ix})$

or $=-64\sin ^7x = {7\choose 0}(\frac{(e^{7ix}-e^{-7ix})}{2i}) - {7\choose 1}(\frac{(e^{5ix} -e^{5ix})}{2i}) + {7\choose 2}(\frac{(e^{3ix} - e^{-3ix})}{2i}) - {7\choose 3}(\frac{(e^{ix} - e^{-ix})}{2i})$

$=1\sin 7x-7\sin 5x+21\sin 3x-35\sin x$

Hence

$\sin^7x=\frac{-1}{64}\sin7x+ \frac{7}{64}\sin5x-\frac{21}{64}\sin3x+ \frac{35}{64}\sin\,x$

What is a trig identity?

A trig identity is an equation that is true for all values of the variables involved. In other words, it is an equation that is always true, no matter what values are substituted for the variables.

Why is proving a trig identity important?

Proving a trig identity is important because it helps to establish the relationship between different trigonometric functions and allows for simplification of more complex expressions.

What is the process for proving a trig identity?

The process for proving a trig identity involves manipulating one side of the equation using algebraic and trigonometric identities until it is equivalent to the other side of the equation.

How do you know if a trig identity is true?

A trig identity is true if both sides of the equation can be simplified to the same expression. This can be verified by substituting different values for the variables and checking if the equation holds true for all values.

What are some common trig identities that can be used to prove this identity?

Some common trig identities that can be used to prove this identity include the double angle, half angle, and sum and difference identities. Additionally, the Pythagorean identities and the power reduction formula may also be helpful in simplifying the expression.

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