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Find the equation of the tangent to the curve y=cos^2 x-sin^2 x at the point where x=pi/4

Find the equation of the tangent to the curve y=cos^2 x-sin^2 x at the point where-example-1
User Niknowj
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3 votes

Answer:


y=-2x+(\pi)/(2)

Explanation:

To find the equation of the tangent line of the curve y = cos²x - sin²x at the point where x = π/4, we first need to find the slope (gradient) of the tangent line at this point.

To find the slope of the tangent line at a specific value of x, we need to evaluate the derivative of the function at that particular value of x.

To differentiate y = cos²x - sin²x, first rewrite the function in terms of cos only by using the cosine double angle identities.


\boxed{\begin{array}{l}\underline{\textsf{Cosine Double Angle Identities}}\\\\\cos 2x=\cos^2 x- \sin^2 x\\\\\cos2x=2 \cos^2 x- 1\\\\\cos 2x=1 - 2 \sin^2 x\\\end{array}}

Therefore, y = cos²x - sin²x can be rewritten as:


y=\cos(2x)

To differentiate y = cos(2x), we can use the chain rule.


\boxed{\begin{array}{c}\underline{\text{Chain Rule for Differentiation}}\\\\\text{If\;\;$y=f(u)$\;\;and\;\;$u=g(x)$\;\;then:}\\\\\frac{\text{d}y}{\text{d}x}=\frac{\text{d}y}{\text{d}u}*\frac{\text{d}u}{\text{d}x}\end{array}}

In this case:


\textsf{Let}\;\;y=\cos(u)\;\;\textsf{where}\;\;u=2x

Differentiate the two parts separately:


y=\cos(u)\implies \frac{\text{d}y}{\text{d}u}=-\sin(u)


u=2x\implies \frac{\text{d}u}{\text{d}x}=2

Now, put everything back into the chain rule formula:


\frac{\text{d}y}{\text{d}x}=-\sin(u) *2


\frac{\text{d}y}{\text{d}x}=-2\sin(u)

Substitute back in u = 2x:


\frac{\text{d}y}{\text{d}x}=-2\sin(2x)

To find the slope of the tangent line at the point where x = π/4, substitute x = π/4 into dy/dx:


\frac{\text{d}y}{\text{d}x}=-2\sin\left(2\cdot (\pi)/(4)\right)


\frac{\text{d}y}{\text{d}x}=-2\sin\left((\pi)/(2)\right)


\frac{\text{d}y}{\text{d}x}=-2(1)


\frac{\text{d}y}{\text{d}x}=-2

Therefore, the slope of the tangent line is m = -2.

Find the corresponding y-coordinate of the point where x = π/4 by substituting x = π/4 into the equation for y:


y=\cos\left(2\cdot (\pi)/(4)\right)


y=\cos\left((\pi)/(2)\right)


y=0

Now, substitute the found slope m = -2 and point (π/4, 0) into the point-slope formula:


\begin{aligned}y-y_1&=m(x-x_1)\\\\y-0&=-2\left(x-(\pi)/(4)\right)\\\\y&=-2x+(\pi)/(2)\end{aligned}

Therefore, the equation of the tangent line of the curve y = cos²x - sin²x at the point where x = π/4 is:


\large\boxed{\boxed{y=-2x+(\pi)/(2)}}

User Catharz
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