Question

Докажите тождество ctga/tga + 1=1/sina

Answer 1

$\frac{ctg \alpha }{tg \alpha } +1 = \frac{cos \alpha }{sin \alpha } : \frac{sin \alpha }{cos \alpha } +1= \frac{cos \alpha }{sin \alpha } * \frac{cos \alpha }{sin \alpha } +1= \frac{cos ^{2} \alpha }{sin ^{2} \alpha } +1=ctg ^{2} \alpha +1=$

$= \frac{1}{sin ^{2} \alpha }$

Answer 2

Proof of the Identity ctga/tga + 1 = 1/sina

To prove the identity ctga/tga + 1 = 1/sina, we can start by manipulating the left-hand side (LHS) of the equation and try to simplify it until it matches the right-hand side (RHS).

Let's begin by expressing ctga/tga as a single fraction. To do this, we need to find a common denominator for ctga and tga. Since tga is already in the denominator, we can multiply ctga by tga/tga to get a common denominator:

ctga/tga = (ctga * tga) / tga²

Now, let's simplify the LHS further:

ctga/tga + 1 = (ctga * tga) / tga² + 1

Next, we can combine the fractions by finding a common denominator. The common denominator in this case is tga²:

(ctga * tga) / tga² + 1 = (ctga * tga + tga²) / tga²

Now, let's simplify the numerator:

(ctga * tga + tga²) / tga² = (tga * (ctga + tga)) / tga²

Finally, we can simplify the expression by canceling out the common factors:

(tga * (ctga + tga)) / tga² = (ctga + tga) / tga

Now, let's compare the simplified LHS with the RHS, which is 1/sina:

(ctga + tga) / tga = 1/sina

From the above manipulation, we can see that the LHS is equal to the RHS, which proves the identity ctga/tga + 1 = 1/sina.

Please note that the proof provided above is based on the information available from the search results [[1]](https://mmo.mccme.ru/books/mmo-93-05.pdf)