Proving the Identity ( sin x cos x frac{sqrt{2}}{2} sinleft(x frac{pi}{4}right) )

Mathematically, the statement ( sin x cos x 1 ) is not accurate as it implies that the product of sine and cosine functions always equals one, which is not true for all values of ( x ). Instead, the correct statement is:

Understanding Identities

An identity in mathematics is an equation that holds true for all values of its variables. The key phrase here is forall ( x in mathbb{R} ), which means the statement is true for all real numbers ( x ).

Existence vs. Universality

The phrase exists ( x in mathbb{R} ) such that ( sin x cos x 1 ) suggests that there are specific values of ( x ) for which this equation holds true. However, it is incorrect to use the notation exists! ( x in mathbb{R} ) such that ( sin x cos x 1 ) because it implies that there is only one such value, which is not the case.

The graph of ( sin x ) and ( cos x ) can be combined into a single function. The product ( sin x cos x ) simplifies into another sine function, demonstrating the trigonometric identity:

A Provable and True Identity

Let's consider the identity: ( forall x in mathbb{R}, sin x cos x frac{sqrt{2}}{2} sinleft(x frac{pi}{4}right) ). We will prove this by transforming ( sin x cos x ) into a form that matches the given identity.

Proving the Identity

Start with the product of sine and cosine:
[sin x cos x] Recognize that ( sin x cos x ) can be expressed in terms of a sine function with a phase shift:
[sin x cos x frac{1}{2} [2 sin x cos x]] Use the double-angle identity for sine, which states that ( 2 sin x cos x sin 2x ). However, we want to transform it using a phase shift:
[sin x cos x frac{1}{2} sin 2x] Use the angle addition formula for sine, (sin(A B) sin A cos B cos A sin B), to express ( sin 2x ) in terms of ( sin left(x frac{pi}{4}right) ). We know that:
[sin 2x sinleft(x xright) sin left(x frac{pi}{4} - frac{pi}{4} xright) sin left(x frac{pi}{4}right) cos left(frac{pi}{4}right) cos left(x frac{pi}{4}right) sin left(frac{pi}{4}right)] Simplify the terms using (cos left(frac{pi}{4}right) sin left(frac{pi}{4}right) frac{sqrt{2}}{2}):
[2 sin x cos x sin left(x frac{pi}{4}right) frac{sqrt{2}}{2} cos left(x frac{pi}{4}right) frac{sqrt{2}}{2} sqrt{2} sin left(x frac{pi}{4}right)] Divide both sides by (sqrt{2}):
[sin x cos x frac{1}{2} sin 2x frac{sqrt{2}}{2} sin left(x frac{pi}{4}right)]

Therefore, we have proven the identity:

[boxed{sin x cos x frac{sqrt{2}}{2} sin left(x frac{pi}{4}right)}]

This confirms that the identity holds for all ( x in mathbb{R} ).

Conclusion

The identity ( sin x cos x frac{sqrt{2}}{2} sin left(x frac{pi}{4}right) ) is a valid and true trigonometric identity. Understanding and proving such identities is crucial in mathematics, particularly in calculus and engineering applications.