sin theta/2cos theta/2=1/2 sine theta show the identiy and work
To prove the identity: sin(theta/2)cos(theta/2) = (1/2)sin(theta), we will use the double-angle formula for sine.
The double-angle formula for sine states that: sin(2x) = 2sin(x)cos(x).
Let's start by substituting theta/2 in place of x in the double-angle formula.
sin(theta) = 2sin(theta/2)cos(theta/2)
Now let's divide both sides of the equation by 2sin(theta/2)cos(theta/2):
sin(theta)/2sin(theta/2)cos(theta/2) = 1
By canceling the common terms on the left-hand side, we get:
1/2cos(theta/2) = 1
Multiplying both sides of the equation by 2 gives us:
cos(theta/2) = 2
Now we can divide both sides of the equation by cos(theta/2) to isolate it:
1 = 2/cos(theta/2)
Recall that sec(x) = 1/cos(x). Applying this, we have:
1 = 2sec(theta/2)
Now, let's take the reciprocal of both sides of the equation:
1/2 = sec(theta/2)
Lastly, we know that secant is the reciprocal of cosine, so let's substitute it back in:
1/2 = 1/cos(theta/2)
Multiplying both sides of the equation by cos(theta/2) gives us:
cos(theta/2) = 1/2
Since cosine is the reciprocal of sine, we can rewrite the equation as:
sin(theta/2) = 1/2
Therefore, sin(theta/2)cos(theta/2) = (1/2)sin(theta).
Hence, the identity sin(theta/2)cos(theta/2) = (1/2)sin(theta) is proven.