Question

((sin a - cos a)^2 - 1)/(tg a - sin a • cos a) = -2ctg^2 a

Answer 1

Применены: формула квадрата разности,  тригонометрические формулы

Answer 2

To prove the identity:

((sin a - cos a)^2 - 1)/(tan a - sin a * cos a) = -2 * cot^2 a

We'll start by working with the left side of the equation and simplifying it step by step:

((sin a - cos a)^2 - 1)/(tan a - sin a * cos a)

First, expand the numerator:

(sin^2 a - 2 * sin a * cos a + cos^2 a - 1)/(tan a - sin a * cos a)

Now, since sin^2 a + cos^2 a = 1 (a trigonometric identity), we can simplify the numerator:

(1 - 2 * sin a * cos a - 1)/(tan a - sin a * cos a)

The "1" and "-1" in the numerator cancel out:

(-2 * sin a * cos a)/(tan a - sin a * cos a)

Next, let's simplify the denominator by expressing tan a as sin a / cos a:

(-2 * sin a * cos a)/((sin a / cos a) - sin a * cos a)

Now, find a common denominator for the terms in the denominator:

(-2 * sin a * cos a * cos a)/((sin a - sin a * cos^2 a))

Now, factor out sin a from the denominator:

(-2 * sin a * cos a * cos a)/(sin a(1 - cos^2 a))

Use the trigonometric identity sin^2 a + cos^2 a = 1 to simplify the denominator further:

(-2 * sin a * cos a * cos a)/(sin a * sin^2 a)

Now, you can cancel out the sin a from the numerator and denominator:

(-2 * cos a * cos a)/(sin a)

Finally, use the identity cos^2 a = 1 - sin^2 a:

(-2 * (1 - sin^2 a))/(sin a)

Now, distribute the -2 to both terms:

(2sin^2 a - 2)/(sin a)

Now, simplify further:

2sin^2 a/sin a - 2/sin a

The sin a in the numerator cancels with one of the sin a terms in the denominator, leaving:

2sin a - 2/sin a

Now, express 2/sin a as 2 * csc a (cosecant):

2sin a - 2csc a

So, the left side of the equation simplifies to:

2sin a - 2csc a

Now, let's work with the right side of the equation, which is -2 * cot^2 a:

-2 * cot^2 a

Since cot a is the reciprocal of tan a, cot a = 1/tan a. So, cot^2 a = (1/tan a)^2 = 1/(tan^2 a).

Now, express tan a as sin a / cos a:

cot^2 a = 1/((sin a / cos a)^2) = 1/(sin^2 a / cos^2 a) = cos^2 a / sin^2 a

Now, the right side of the equation becomes:

-2 * cos^2 a / sin^2 a

So, we have the left side as 2sin a - 2csc a and the right side as -2cos^2 a / sin^2 a. We can see that these two expressions are not equivalent, so the initial identity is not valid.

The identity you provided, ((sin a - cos a)^2 - 1)/(tan a - sin a * cos a) = -2 * cot^2 a, is not true.