Find out the answer to "How cos2y dy become cos 2y/2"

raghumanu534

Answered question

2022-04-14

How cos2y dy become cos 2y/2

user_27qwe

Skilled2023-04-27Added 375 answers

To understand how

\cos (2 y) d y

becomes

\frac{1}{2} \cos (2 y)

, we can use a trigonometric identity.
Recall the double angle formula for cosine:

\cos (2 θ) = 2 \cos^{2} (θ) - 1

Letting

θ = y

, we have:

\cos (2 y) = 2 \cos^{2} (y) - 1

Solving for

\cos^{2} (y)

, we get:

\cos^{2} (y) = \frac{1}{2} (\cos (2 y) + 1)

Now, we can rewrite

\cos (2 y) d y

as:

\cos (2 y) d y = \cos (2 y) \cdot 1 \cdot d y = \cos (2 y) \cdot \frac{1}{2} \cdot 2 d y

Substituting in the expression for

\cos^{2} (y)

, we have:

\cos (2 y) d y = \cos (2 y) \cdot \frac{1}{2} \cdot 2 d y = \frac{1}{2} (2 \cos^{2} (y)) d y = \frac{1}{2} (\cos (2 y) + 1) d y

Therefore, we have shown that

\cos (2 y) d y

can be written as

\frac{1}{2} (\cos (2 y) + 1) d y

.
To simplify this expression further, note that

\frac{1}{2} (\cos (2 y) + 1)

can be written as

\frac{1}{2} \cos (2 y) + \frac{1}{2}

.
So,

\cos (2 y) d y = \frac{1}{2} \cos (2 y) d y + \frac{1}{2} d y

Subtracting

\frac{1}{2} \cos (2 y) d y

from both sides, we get:

\frac{1}{2} \cos (2 y) d y = \frac{1}{2} d y

Dividing both sides by

\frac{1}{2}

, we get:

\cos (2 y) d y = d y

Therefore, we have shown that

\cos (2 y) d y

is equivalent to

d y

, which implies that

\int \cos (2 y) d y = \frac{1}{2} \sin (2 y) + C

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