how to solve sin 4x= cosx(4 sinx - 8 sin ^3x)
Can you check if the question requires to solve for x or to prove the identity? I believe it is the latter.
Work on the right-hand-side:
cos(x)(4sin(x)-8sin³(x))
=4cos(x)sin(x)(1-2sin²(x))... factorization
=2sin(2x)(1-2sin²(x))... double angle formula
=2sin(2x)(1-2(1/2)(cos(0)-cos(2x)))... products formula
=2sin(2x)(1-1+cos(2x))
=2sin(2x)cos(2x)
=sin(4x) ... double angle formula
= left-hand-side
is to prove an identity
Done! :)
prove that cos thete + sec thete is equal 2 1
To solve the equation sin(4x) = cos(x) * (4sin(x) - 8sin^3(x), we can use trigonometric identities to simplify the expression.
Let's start by applying the double-angle identity for sine: sin(2θ) = 2sin(θ)cos(θ). We can express sin(4x) as sin(2 * 2x).
sin(4x) = sin(2 * 2x) = 2sin(2x)cos(2x)
Next, we'll apply the double-angle identity for cosine: cos(2θ) = cos^2(θ) - sin^2(θ). We can express cos(2x) as cos(x)^2 - sin(x)^2.
sin(4x) = 2sin(2x)(cos(x)^2 - sin(x)^2)
Now, we'll apply the double-angle identity for sine again to simplify sin(2x):
sin(2x) = 2sin(x)cos(x)
Substituting this back into the equation, we have:
2sin(2x)(cos(x)^2 - sin(x)^2) = cos(x) * (4sin(x) - 8sin^3(x))
Simplifying further, we get:
2(2sin(x)cos(x))(cos(x)^2 - sin(x)^2) = cos(x) * (4sin(x) - 8sin^3(x))
4sin(x)cos^3(x) - 4sin^3(x)cos(x) = 4sin(x)cos(x) - 8sin^3(x)cos(x)
Now, we can factor out sin(x)cos(x):
sin(x)cos(x)(4cos^2(x) - 4sin^2(x) - 4 + 8sin^2(x)) = 0
sin(x)cos(x)(4cos^2(x) + 4sin^2(x) - 4) = 0
sin(x)cos(x)(4(cos^2(x) + sin^2(x)) - 4) = 0
sin(x)cos(x)(4 - 4) = 0
sin(x)cos(x) * 0 = 0
Therefore, either sin(x) = 0, cos(x) = 0, or both sin(x) and cos(x) are zero simultaneously.
To solve for x, we examine each case:
1. When sin(x) = 0, x can take values of 0, π, 2π, 3π, etc., since sin(0) = sin(π) = sin(2π) = sin(3π) = 0.
2. When cos(x) = 0, x can take values of π/2, 3π/2, 5π/2, etc., since cos(π/2) = cos(3π/2) = cos(5π/2) = 0.
3. When sin(x) = 0 and cos(x) = 0 simultaneously, x can take values of π/2, 3π/2, 5π/2, etc., as both sine and cosine are zero at these points.
These are the solutions for the equation sin(4x) = cos(x) * (4sin(x) - 8sin^3(x)).