Question

Cos a + ctg a / 1 + sin a

Answer 1

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Answer 2

To simplify the expression cos(a) + cot(a) / (1 + sin(a)), we can start by simplifying the cotangent term using its definition:

cot(a) = 1 / tan(a)

Substituting this back into the expression, we have:

cos(a) + (1 / tan(a)) / (1 + sin(a))

Next, let's simplify the fraction by multiplying the numerator and denominator by tan(a):

cos(a) * tan(a) + 1 / (tan(a) * (1 + sin(a)))

Now, we can simplify the numerator by using the identity:

cos(a) * tan(a) = sin(a)

Substituting this back into the expression, we have:

sin(a) + 1 / (tan(a) * (1 + sin(a)))

Next, we can simplify the denominator by using the identity:

tan(a) = sin(a) / cos(a)

Substituting this into the expression, we have:

sin(a) + 1 / ((sin(a) / cos(a)) * (1 + sin(a)))

Simplifying further, we get:

sin(a) + 1 / (sin(a) * (1 + sin(a)) / cos(a))

To simplify the fraction, we can multiply by the reciprocal of the denominator:

sin(a) + cos(a) / (sin(a) * (1 + sin(a)))

Now, we can multiply out the denominator:

sin(a) + cos(a) / (sin(a) + sin^2(a))

Finally, we can combine the terms in the numerator:

(sin(a) + cos(a)) / (sin(a) + sin^2(a))

So, the simplified expression is:

(sin(a) + cos(a)) / (sin(a) + sin^2(a))