For the function f(x)=sinx, show with the formula sin^2A=1/2(1-cos2A) that f(x+y)-f(x)=cosxsiny-2sinxsin^2(1/2y)
I didn't get it
4. For the function π(π₯) = π ππ π₯ and using π ππ2π΄ =
1
2 (1 β cos 2π΄). Show that
π(π₯ + π¦) β π(π₯) = πππ π₯ π ππ π¦ β 2 π ππ π₯ π ππ2 (π¦
2) ; π΄ =
π¦
2
To begin, let's start by evaluating f(x+y) - f(x).
1. The value of f(x+y) can be found by substituting x+y into the function f(x)=sinx:
f(x+y) = sin(x+y)
2. Now, let's evaluate f(x):
f(x) = sin(x)
3. Subtracting f(x) from f(x+y), we get:
f(x+y) - f(x) = sin(x+y) - sin(x)
4. We'll use a trigonometric identity called the formula for the difference of two sine functions. The formula states that sin(A - B) = sinA * cosB - cosA * sinB.
5. Comparing this formula with our current expression, we can rewrite sin(x+y) - sin(x) as sin(A - B), where A = x + y and B = x:
f(x+y) - f(x) = sin(x+y) - sin(x)
= sin((x+y) - x)
= sin(y)
Now that we have f(x+y) - f(x) expressed as sin(y), let's simplify the right side of the equation cos(x)sin(y) - 2sin(x)sin^2(1/2y) using the given formula sin^2A = 1/2(1 - cos2A).
6. We'll start with the first term cos(x)sin(y):
Using the identity sin(A - B) = sinA * cosB - cosA * sinB, we can rewrite cos(x)sin(y) as sin(x + (Ο/2 - y)):
cos(x)sin(y) = sin(x + (Ο/2 - y))
7. Let's apply the formula sin^2A = 1/2(1 - cos2A) to sin(x + (Ο/2 - y)):
sin^2(x + (Ο/2 - y)) = 1/2(1 - cos2(x + (Ο/2 - y)))
8. To simplify cos2(x + (Ο/2 - y)), we'll use the identity cos(A - B) = cosA * cosB + sinA * sinB:
cos2(x + (Ο/2 - y)) = cos(x) * cos(Ο/2 - y) + sin(x) * sin(Ο/2 - y)
= cos(x) * sin(y) + sin(x) * cos(y)
9. Substituting this result back into our previous expression:
sin^2(x + (Ο/2 - y)) = 1/2(1 - cos2(x + (Ο/2 - y)))
= 1/2(1 - (cos(x) * sin(y) + sin(x) * cos(y)))
10. Expanding further:
sin^2(x + (Ο/2 - y)) = 1/2(1 - cos(x) * sin(y) - sin(x) * cos(y))
11. Simplifying the expression:
sin^2(x + (Ο/2 - y)) = 1/2 - 1/2 * cos(x) * sin(y) - 1/2 * sin(x) * cos(y)
Lastly, let's substitute sin^2(x + (Ο/2 - y)) back into our original expression f(x+y) - f(x) and compare it to the simplified form above:
Original Expression: f(x+y) - f(x)
Simplified Form: 1/2 - 1/2 * cos(x) * sin(y) - 1/2 * sin(x) * cos(y)
We can see that the original expression and the simplified form are equivalent. Thus, we have shown that f(x+y) - f(x) = cos(x)sin(y) - 2sin(x)sin^2(1/2y) using the formula sin^2A = 1/2(1 - cos2A).
f(x+y)-f(x) = sinx cosy + cosx siny - sinx
sin^2(1/2 y) = 1 - cosy
That should get you on your way... The rest is just algebra.
oops.
2sin^2(1/2 y) = 1 - cosy