Question

Упростить выражение cos3a/cosa - sin3a/sina

Answer 1

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Answer 2

I can help you with math problems and other topics. I will try to simplify the expression cos3a/cosa - sin3a/sina for you.

To simplify this expression, we can use the following trigonometric identities:

- cos(a + b) = cos(a)cos(b) - sin(a)sin(b) - sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

Using these identities, we can rewrite cos3a and sin3a as follows:

- cos3a = cos(2a + a) = cos(2a)cos(a) - sin(2a)sin(a) - sin3a = sin(2a + a) = sin(2a)cos(a) + cos(2a)sin(a)

We can also use the double angle formulas for cos(2a) and sin(2a):

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

Substituting these formulas into cos3a and sin3a, we get:

- cos3a = (cos^2(a) - sin^2(a))cos(a) - 2sin(a)cos(a)sin(a) - sin3a = 2sin(a)cos(a)(cos(a) + sin(a))

Now we can divide cos3a by cosa and sin3a by sina, and simplify the fractions:

- cos3a/cosa = (cos^2(a) - sin^2(a)) - 2sin(a)cos(a)tan(a) - sin3a/sina = 2cos(a)(1 + tan(a))

Finally, we can subtract sin3a/sina from cos3a/cosa, and simplify the result:

- cos3a/cosa - sin3a/sina = (cos^2(a) - sin^2(a)) - 2sin(a)cos(a)tan(a) - 2cos(a)(1 + tan(a)) - cos3a/cosa - sin3a/sina = cos^2(a) - 3sin^2(a) - 4sin(a)cos(a)tan(a) - 2cos(a)

This is the simplified expression. You can also check the answer using [Microsoft Math Solver](https://mathsolver.microsoft.com/en/solve-problem/%60frac%20%7B%20%60cos%20A%20-%20%60cos%203%20A%20%7D%20%7B%20%60cos%20A%20%7D%20%2B%20%60frac%20%7B%20%60sin%20A%20%2B%20%60sin%203%20A%20%7D%20%7B%20%60sin%20A%20%7D%20%3D) or other online tools. I hope this helps.