Question

(cos 2a - sin 2a)/cos4a

Answer 1

$\frac{cos2 \alpha -sin2 \alpha }{cos4 \alpha }= \frac{cos2 \alpha -sin2 \alpha }{cos^22 \alpha -sin^22 \alpha }= \frac{cos2 \alpha -sin2 \alpha }{(cos2 \alpha -sin2 \alpha )(cos2 \alpha +sin2 \alpha) }=\\= \frac{1}{cos2 \alpha +sin2 \alpha }$
УДАЧИ ВАМ ВО ВСЁМ)))!

Answer 2

Если надо, то можно ещё домножить на (-1), чтобы получить формулу квадрата суммы в знаменателе

Answer 3

To simplify the expression (cos 2a - sin 2a)/cos 4a, we can use the trigonometric identity:

cos(2a) = cos^2(a) - sin^2(a)

Also, we know that:

sin(2a) = 2 * sin(a) * cos(a)

Now, let's substitute these identities into the expression:

(cos 2a - sin 2a)/cos 4a = (cos^2(a) - sin^2(a) - 2 * sin(a) * cos(a))/cos(2 * 2a)

Now, let's further simplify the numerator:

cos^2(a) - sin^2(a) = cos^2(a) - (1 - cos^2(a)) [Using the identity: sin^2(a) = 1 - cos^2(a)] = cos^2(a) - 1 + cos^2(a) = 2 * cos^2(a) - 1

Substitute back into the expression:

= (2 * cos^2(a) - 1 - 2 * sin(a) * cos(a)) / cos(2 * 2a)

Next, let's use the double angle identity for cosine:

cos(2 * 2a) = cos^2(2a) - sin^2(2a)

We already know:

cos(2a) = cos^2(a) - sin^2(a) sin(2a) = 2 * sin(a) * cos(a)

Using these, we can find cos(2 * 2a):

cos(2 * 2a) = cos^2(2a) - sin^2(2a) = (cos(2a))^2 - (sin(2a))^2 = (cos^2(a) - sin^2(a))^2 - (2 * sin(a) * cos(a))^2 = (cos^2(a) - sin^2(a))^2 - 4 * (sin(a) * cos(a))^2 = (cos^2(a) - sin^2(a))^2 - 4 * (sin^2(a) * cos^2(a))

Now, we can substitute this into our expression:

= (2 * cos^2(a) - 1 - 2 * sin(a) * cos(a)) / [ (cos^2(a) - sin^2(a))^2 - 4 * (sin^2(a) * cos^2(a)) ]

We can further simplify the denominator using the Pythagorean identity:

cos^2(a) + sin^2(a) = 1

So, cos^2(a) = 1 - sin^2(a)

Substitute this back into the denominator:

= (2 * cos^2(a) - 1 - 2 * sin(a) * cos(a)) / [ ((1 - sin^2(a)) - sin^2(a))^2 - 4 * (sin^2(a) * (1 - sin^2(a))) ]

= (2 * cos^2(a) - 1 - 2 * sin(a) * cos(a)) / [ (1 - 2 * sin^2(a))^2 - 4 * sin^2(a) * (1 - sin^2(a)) ]

Now, we can further simplify the numerator:

= [2 * (cos^2(a) - sin(a) * cos(a)) - 1] / [ (1 - 2 * sin^2(a))^2 - 4 * sin^2(a) * (1 - sin^2(a)) ]

= [2 * cos(a) * (cos(a) - sin(a)) - 1] / [ (1 - 2 * sin^2(a))^2 - 4 * sin^2(a) * (1 - sin^2(a)) ]

That's the final simplified expression for (cos 2a - sin 2a)/cos 4a.