I need help with verifying these trig identities:
1) sin4x = 4sinxcos - 8sin^3 x cos x
2) cos3x = cos^3 x - 3sin^2 x cos x
sin4x = 4sinxcos - 8sin^3x cosx
sin4x = 4sinx*cosx(1-2sin^2x)
sin4x = 4sinx*cosx(cos2x)
sin4x = 2(2sinx*cosx)(cos2x)
sin4x = 2(sin2x)(cos2x)
sin4x = sin(2*2x)
sin4x = sin4x
cos3x
cos(x+2x)
cosx*cos2x - sinx*sin2x
cosx(1-2sin^2x) - sinx(2sinx*cosx)
cosx(1 - 2sin^2x - 2sin^2x)
cosx(1-4sin^2x)
cosx(cos^2x+sin^2x-4sin^2x)
cosx(cos^2x-3sin^2x)
cos^3x - 3sin^2x cosx
To verify the given trigonometric identities, we will use various trigonometric identities and properties. Let's start with the first identity:
1) sin4x = 4sinxcosx - 8sin^3xcosx
To verify this identity, we need to transform one side of the equation to match the other side. Here's how we can do it step by step:
Step 1: Express sin4x using the double-angle identity for sine:
sin4x = 2sin2xcos2x
Step 2: Express sin2x using the double-angle identity for sine:
sin2x = 2sinxcosx
Step 3: Substitute sin2x back into sin4x:
sin4x = 2(2sinxcosx)cos2x
Step 4: Express cos2x using the identity: cos2x = 1 - sin^2x
sin4x = 2(2sinxcosx)(1 - sin^2x)
Step 5: Simplify:
sin4x = 4sinxcosx - 4sin^3xcosx
Step 6: Factor out 4sinxcosx from the right side:
sin4x = 4sinxcosx(1 - sin^2x)
Step 7: Use the Pythagorean identity sin^2x + cos^2x = 1 to replace 1 - sin^2x:
sin4x = 4sinxcosx(cos^2x)
Step 8: Finally, we have:
sin4x = 4sinxcosx - 8sin^3x(cosx)
As both sides of the equation match, we have verified the first trigonometric identity.
Now let's move on to the second identity:
2) cos3x = cos^3x - 3sin^2x cosx
Again, we need to transform one side of the equation to match the other side. Here's the step-by-step process:
Step 1: Express cos3x using the triple-angle identity for cosine:
cos3x = (4cos^3x)-3cosx
Step 2: Simplify the right side of the equation:
cos3x = 4cos^3x - 3cosx
Step 3: Use the identity sin^2x = 1 - cos^2x to replace cos^3x:
cos3x = 4(1 - sin^2x) - 3cosx
Step 4: Distribute 4 across the expression:
cos3x = 4 - 4sin^2x - 3cosx
Step 5: Use the identity sin^2x = 1 - cos^2x to replace sin^2x:
cos3x = 4 - 4(1 - cos^2x) - 3cosx
Step 6: Simplify and distribute -4:
cos3x = 4 - 4 + 4cos^2x - 3cosx
Step 7: Combine like terms:
cos3x = 4cos^2x - 3cosx
As both sides of the equation match, we have verified the second trigonometric identity.
That's how you verify these trig identities. By applying various trigonometric identities and simplification techniques, you can transform both sides of the equation to match each other.