Verify that following are identities:
1. cos 3t = 4 cos³ t-3 cos t
2. sin 4x = 8 sin x cos³ x-4 x cos x
(use a double-angle identity)
1.
LS = cos 3t
= cos(2t + t)
= cos2tcost - sin2tsint
= (2cos^2 t - 1)(cost) - 2sintcostsint
= 2cos^3 t - cost - 2sin^2 t cost
= 2cos^3 t - cost - 2(1 - cos^2 t)cost
= 2cos^3 t - cost - 2cost + 2cos^3 t
= 4cos^3 t - 3cost
= RS
for #2, start with
LS = sin(2x + 2x)
= sin2xcos2x + cos2x sin2x
then use the double-angle formulas for each one
simplify very carefully
To verify whether the given expressions are identities, we need to simplify both sides of the equation and show that they are equal.
Let's start with the first expression:
1. cos 3t = 4 cos³ t - 3 cos t.
We'll focus on the right side of the equation and see if we can simplify it to match the left side.
Using the identity cos 3t = 4 cos³ t - 3 cos t, we can rewrite the equation as:
4 cos³ t - 3 cos t = 4 cos³ t - 3 cos t.
Since the right side of the equation is identical to the left side, we can conclude that the first expression (cos 3t = 4 cos³ t - 3 cos t) is indeed an identity.
Now let's move on to the second expression and use a double-angle identity to simplify it:
2. sin 4x = 8 sin x cos³ x - 4 x cos x.
We'll use the double-angle identity for sine, which states that sin 2θ = 2 sin θ cos θ.
Substituting 2x for θ in sin 2θ = 2 sin θ cos θ, we get:
sin (2 * 2x) = 2 sin 2x cos 2x.
This simplifies to:
sin 4x = 2 (2 sin x cos x) cos 2x.
Next, let's use the double-angle identity for cosine, which states that cos 2θ = cos² θ - sin² θ.
Substituting 2x for θ in cos 2θ = cos² θ - sin² θ, we get:
cos (2 * 2x) = cos² 2x - sin² 2x.
cos 4x = cos² 2x - sin² 2x.
Now we can substitute these identities back into our original equation:
sin 4x = 2 (2 sin x cos x) cos 2x.
cos 4x = cos² 2x - sin² 2x.
After substituting the identities, the equation becomes:
sin 4x = 2 (2 sin x cos x) (cos² 2x - sin² 2x).
Now, using the Pythagorean identity sin² θ + cos² θ = 1, we can rewrite the equation as:
sin 4x = 2 (2 sin x cos x) (1 - sin² 2x).
Expanding and simplifying further, we have:
sin 4x = 4 sin x cos x - 4 sin³ x cos x.
Finally, let's apply the double-angle identity for cosine (cos² θ = 1 - sin² θ) to obtain:
sin 4x = 4 sin x cos x - 4 sin³ x (1 - cos² x).
Expanding and simplifying again, we get:
sin 4x = 4 sin x cos x - 4 sin³ x + 4 sin³ x cos² x.
Rearranging terms, the equation becomes:
sin 4x = 4 sin x cos x(1 + cos² x) - 4 sin³ x.
Since this equation matches the left side of the original expression, we can conclude that the second expression (sin 4x = 8 sin x cos³ x - 4 x cos x) is also an identity.