How cos2y dy become cos 2y/2

raghumanu534

raghumanu534

Answered question

2022-04-14

How cos2y dy become cos 2y/2

Answer & Explanation

user_27qwe

user_27qwe

Skilled2023-04-27Added 375 answers

To understand how cos(2y)dy becomes 12cos(2y), we can use a trigonometric identity.
Recall the double angle formula for cosine:
cos(2θ)=2cos2(θ)1
Letting θ=y, we have:
cos(2y)=2cos2(y)1
Solving for cos2(y), we get:
cos2(y)=12(cos(2y)+1)
Now, we can rewrite cos(2y)dy as:
cos(2y)dy=cos(2y)·1·dy=cos(2y)·12·2dy
Substituting in the expression for cos2(y), we have:
cos(2y)dy=cos(2y)·12·2dy=12(2cos2(y))dy=12(cos(2y)+1)dy
Therefore, we have shown that cos(2y)dy can be written as 12(cos(2y)+1)dy.
To simplify this expression further, note that 12(cos(2y)+1) can be written as 12cos(2y)+12.
So, cos(2y)dy=12cos(2y)dy+12dy
Subtracting 12cos(2y)dy from both sides, we get:
12cos(2y)dy=12dy
Dividing both sides by 12, we get:
cos(2y)dy=dy
Therefore, we have shown that cos(2y)dy is equivalent to dy, which implies that cos(2y)dy=12sin(2y)+C.

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