Question

Упростить : Cos 72 + cos 18

Answer 1

$\cos72^o+cos18^0=\cos\left(90^0-18^o\right)+\cos\left(0^0+18^0\right)=\\ =\cos90^0\cdot\cos18^0+\sin90^0\cdot\sin18^0+\cos0^0\cdot\cos18^0-sin0^0\cdot\sin18^0=\\ =0\cdot\cos18^0+1\cdot\sin18^0+1\cdot\cos18^0-0\cdot\sin18^0=\\ =sin18^0+\cos18^0$

Answer 2

To simplify the expression "cos 72 + cos 18," we can use trigonometric identities. One of the useful identities is:

cos (A + B) = cos A * cos B - sin A * sin B

Let's use this identity to simplify the given expression:

cos 72 + cos 18

We can rewrite 72 as the sum of 18 and 54:

cos (18 + 54) + cos 18

Now, using the cosine addition identity, we get:

cos 18 * cos 54 - sin 18 * sin 54 + cos 18

Since cos (90 - x) = sin x, we can rewrite cos 54 as sin 36:

cos 18 * sin 36 - sin 18 * sin 54 + cos 18

Now, let's use the identity sin (A + B) = sin A * cos B + cos A * sin B:

sin (18 + 54) + cos 18

sin 72 + cos 18

Finally, the angle 72 degrees corresponds to cos 72 = cos (90 - 18) = sin 18. Therefore:

sin 72 + cos 18 = sin 18 + cos 18

And that's the simplified form of the expression.