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leopard
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Why is [tex]cos^2 x = \frac{1}{2} + \frac{cos(2x)}{2}[/tex]
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leopard said:Why is [tex]cos^2 x = \frac{1}{2} + \frac{cos(2x)}{2}[/tex]
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A trigonometric identity is an equation involving trigonometric functions that is true for all values of the variables involved. In other words, it is a mathematical statement that is always true, regardless of the specific values of the angles involved.
This trigonometric identity can be used to simplify and solve equations involving trigonometric functions. It is particularly useful in solving problems involving trigonometric identities, as it allows us to rewrite one function in terms of another and ultimately simplify the equation.
The number 1/2 represents the cosine of the angle x when squared. This means that if we square the cosine of any angle, the result will always be 1/2. This is a key component of the identity and is derived from the Pythagorean identity for cosine.
The term cos(2x)/2 represents the double angle formula for cosine. This means that if we double the angle x, the cosine of that angle will be divided by 2. This term allows us to rewrite the identity in terms of a single angle, rather than a squared cosine, which can be more useful in certain equations.
To verify the identity, you can substitute different values for x and evaluate both sides of the equation. If the resulting values are equal, the identity is confirmed to be true. This can also be done algebraically by manipulating the equation and using known trigonometric identities to show that both sides are equal.