Find the value of x: sin(x)=cos(x) for x in [0, π/2].
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Step-by-Step Solution
Step 1: Rewrite the equation
The given equation is sin(x)=cos(x). To solve for x, we can divide both sides by cos(x), provided that cos(x)=0. This will allow us to express the equation in terms of tan(x).
Step 2: Divide by cos(x)
Dividing both sides of the equation by cos(x) gives us cos(x)sin(x)=1. We know that cos(x)sin(x) is equal to tan(x).
Step 3: Express in terms of tan(x)
Substituting tan(x) for cos(x)sin(x), the equation simplifies to tan(x)=1. Now we need to find the value of x for which the tangent is 1.
Step 4: Find the value of x
We are looking for the angle x in the interval [0,π/2] whose tangent is 1. We know that tan(π/4)=1. Therefore, x=π/4. This value lies within the given interval.