Question

Sina/1+cosa-sina/1-cosa=-2ctga

Answer 1

Смотри..................

Answer 2

The equation you've provided is a trigonometric equation. To solve it, we can manipulate it step by step. Here's what the equation looks like:

(sinα / (1 + cosα)) - (sinα / (1 - cosα)) = -2ctgα

Where ctgα represents the cotangent of α.

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

(sinα(1 - cosα) - sinα(1 + cosα)) / ((1 + cosα)(1 - cosα)) = -2ctgα

Now, simplify the equation further:

(sinα - sinα * cosα - sinα - sinα * cosα) / (1 - cos²α) = -2ctgα

Combine like terms in the numerator:

-2sinα(1 - cosα) / (1 - cos²α) = -2ctgα

Now, we can use trigonometric identities to simplify further. The identity we'll use is:

ctgα = 1/tanα

Now, we have:

-2sinα(1 - cosα) / (1 - cos²α) = -2/tanα

Now, we can simplify the right side of the equation by using the identity tanα = sinα/cosα:

-2sinα(1 - cosα) / (1 - cos²α) = -2(sinα/cosα)

Now, let's focus on the left side of the equation and use the Pythagorean identity for sin²α + cos²α = 1:

1 - cos²α = sin²α

Substitute this into the left side:

-2sinα(1 - cosα) / sin²α

Now, you can cancel a factor of sinα from the numerator and denominator:

-2(1 - cosα) / sinα

Now, the equation becomes:

-2(1 - cosα) / sinα = -2sinα/cosα

Now, cross-multiply:

-2(1 - cosα)cosα = -2sin²α

Distribute the -2 on the left side:

-2cosα + 2cos²α = -2sin²α

Now, simplify:

2sin²α + 2cos²α - 2cosα = 0

Use the trigonometric identity sin²α + cos²α = 1:

2(1 - cosα) - 2cosα = 0

Now, distribute the 2:

2 - 2cosα - 2cosα = 0

Combine like terms:

2 - 4cosα = 0

Now, isolate cosα:

4cosα = 2

cosα = 2/4

cosα = 1/2

So, the solution to the equation is cosα = 1/2.