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anemone
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Prove that $\sin^7 x=\dfrac{35\sin x-21\sin 3x+7\sin 5x-\sin 7x}{64}$.
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.
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.
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.
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.
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.