Question

Ctg x + sin x / 1+cos x​

Answer 1

Ответ:

$\frac{1}{sinx}$

Объяснение:

$\frac{cosx}{sinx} +\frac{sinx}{1+cosx} = \frac{(1+cosx)*cosx+sinx*sinx}{sin*(1+cosx)} = \frac{cosx+cos^2x+sin^2x}{sin*(1+cosx)} = \frac{1+cosx}{sin*(1+cosx)} = \frac{1}{sinx}$

Используется основное тригонометрическое равенство:

sin^2x + cos^2x = 1

Answer 2

To simplify the expression (ctg x + sin x) / (1 + cos x), we can first rewrite the trigonometric functions in terms of basic trigonometric identities.

1. cotangent (ctg x) is the reciprocal of tangent (tan x): ctg x = 1 / tan x

2. cosine (cos x) is the reciprocal of secant (sec x): cos x = 1 / sec x

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

(ctg x + sin x) / (1 + cos x) = (1 / tan x + sin x) / (1 + 1 / sec x) = (1 / tan x + sin x) / (1 + sec x)

Next, let's find a common denominator for the two fractions:

Common denominator = tan x * sec x

Now, adjust the numerators accordingly:

= [(1 + sin x * tan x) / (tan x * sec x)] / [(1 * sec x + sec x) / (tan x * sec x)]

Now, simplify the expression further:

= (1 + sin x * tan x) / (sec x + sec x * tan x)

Since tan x = sin x / cos x and sec x = 1 / cos x:

= (1 + sin x * (sin x / cos x)) / (1 / cos x + (1 / cos x) * (sin x / cos x))

= (1 + sin^2 x / cos x) / (1 / cos x + sin x / cos^2 x)

Now, find the common denominator again:

Common denominator = cos x * cos^2 x = cos^3 x

= [(1 + sin^2 x / cos x) / cos^3 x] / [(1 + sin x) / cos^2 x]

Invert the second fraction and multiply:

= (1 + sin^2 x / cos x) / cos^3 x * cos^2 x / (1 + sin x)

= (1 + sin^2 x / cos x) / (cos x * (1 + sin x))

Now, recall the Pythagorean identity: sin^2 x + cos^2 x = 1

This means sin^2 x = 1 - cos^2 x. Substituting it into the expression:

= (1 + (1 - cos^2 x) / cos x) / (cos x * (1 + sin x))

= (1 + (1 - cos^2 x) / cos x) / (cos x + cos x * sin x)

= (1 + (1 / cos x - cos^2 x / cos x)) / (cos x + cos x * sin x)

= (1 + (1 / cos x - cos x)) / (cos x + cos x * sin x)

= (1 - cos x + 1 / cos x) / (cos x * (1 + sin x))

Finally, the simplified expression is:

(1 - cos x + 1 / cos x) / (cos x * (1 + sin x))