Question

((1+tnA)(1-ctgA))^2=1/(sin^2Acos^2A)-4

Answer 1

Ответ:

((1+tnA)(1-ctgA))^2=1/(sin^2Acos^2A)-4=5

Answer 2

Let's simplify the given expression step by step:

Starting with the left side of the equation:

((1 + tnA)(1 - ctgA))^2

First, we need to recall some trigonometric identities:

1. tan(A) = sin(A) / cos(A)
2. cot(A) = 1 / tan(A) = cos(A) / sin(A)

Now, we can rewrite the expression using these identities:

((1 + sin(A)/cos(A))(1 - cos(A)/sin(A)))^2

Now, let's simplify further:

((cos(A) + sin(A))/cos(A))(1 - cos(A)/sin(A)))^2

Now, let's combine the fractions:

((cos(A) + sin(A))(sin(A) - cos(A))/ (cos(A)sin(A)))^2

Now, expand the square:

((cos(A) + sin(A))(sin(A) - cos(A))^2/ (cos(A)sin(A))^2

Now, let's simplify the right side of the equation:

1/(sin^2(A)cos^2(A)) - 4

Now, we need to find a common denominator for the fractions in the equation:

(sin^2(A)cos^2(A)/(sin^2(A)cos^2(A)) - 4(sin^2(A)cos^2(A)/(sin^2(A)cos^2(A)))

Now, combine the fractions:

(1 - 4sin^2(A)cos^2(A)/(sin^2(A)cos^2(A)))

Now, simplify the right side:

1 - 4

So, the right side simplifies to -3.

Now, the equation becomes:

((cos(A) + sin(A))(sin(A) - cos(A))^2/ (cos(A)sin(A))^2 = -3

We can further simplify this expression, but it doesn't seem to lead to a simple, obvious identity. The equation represents a trigonometric relationship, but it doesn't simplify to a well-known trigonometric identity. If you have a specific question or context related to this equation, please provide more details, and I'll do my best to assist you further.