Certainly! Let's go through the steps to find the slope of the tangent to the function [tex]\( y = \cos^2 x - \sin^2 x \)[/tex] at the point [tex]\( x = \frac{3\pi}{4} \)[/tex].
### Step 1: Differentiate the function
To find the slope of the tangent line, we first need to compute the derivative [tex]\( \frac{dy}{dx} \)[/tex] of the function [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex].
The function we have is:
[tex]\[ y = \cos^2 x - \sin^2 x \][/tex]
Using the chain rule, we differentiate each term separately.
For [tex]\( \cos^2 x \)[/tex], we use the chain rule:
[tex]\[ \frac{d}{dx} (\cos^2 x) = 2 \cos x \cdot (-\sin x) = -2 \cos x \sin x \][/tex]
For [tex]\( \sin^2 x \)[/tex], we use the chain rule:
[tex]\[ \frac{d}{dx} (\sin^2 x) = 2 \sin x \cdot \cos x = 2 \sin x \cos x \][/tex]
Combining these results, the derivative of [tex]\( y = \cos^2 x - \sin^2 x \)[/tex] is:
[tex]\[ \frac{dy}{dx} = -2 \cos x \sin x - 2 \sin x \cos x \][/tex]
[tex]\[ \frac{dy}{dx} = -4 \cos x \sin x \][/tex]
### Step 2: Evaluate the derivative at [tex]\( x = \frac{3\pi}{4} \)[/tex]
Now, we need to evaluate the derivative [tex]\( \frac{dy}{dx} \)[/tex] at the specific point [tex]\( x = \frac{3\pi}{4} \)[/tex].
First, determine the values of [tex]\( \cos \left( \frac{3\pi}{4} \right) \)[/tex] and [tex]\( \sin \left( \frac{3\pi}{4} \right) \)[/tex]:
[tex]\[ \cos \left( \frac{3\pi}{4} \right) = -\frac{\sqrt{2}}{2} \][/tex]
[tex]\[ \sin \left( \frac{3\pi}{4} \right) = \frac{\sqrt{2}}{2} \][/tex]
Now substitute these values into the derivative:
[tex]\[ \frac{dy}{dx} \bigg|_{x = \frac{3\pi}{4}} = -4 \cos \left( \frac{3\pi}{4} \right) \sin \left( \frac{3\pi}{4} \right) \][/tex]
[tex]\[ = -4 \left( -\frac{\sqrt{2}}{2} \right) \left( \frac{\sqrt{2}}{2} \right) \][/tex]
[tex]\[ = -4 \left( -\frac{2}{4} \right) \][/tex]
[tex]\[ = -4 \left( -\frac{1}{2} \right) \][/tex]
[tex]\[ = 2 \][/tex]
### Final Answer
Therefore, the slope of the tangent to the curve [tex]\( y = \cos^2 x - \sin^2 x \)[/tex] at the point where [tex]\( x = \frac{3\pi}{4} \)[/tex] is:
[tex]\[ \boxed{2} \][/tex]
