Question

Упростить выражение (1/sin^2(x))-1

Answer 1

$\frac{1}{ sin^{2}x } - 1^{ (sin^{2}x } = \frac{1- sin^{2} x}{ sin^{2}x } = \frac{ cos^{2}x }{ sin^{2}x } = ctg^{2} x$

Answer 2

To simplify the expression (1/sin^2(x)) - 1, let's first find a common denominator and then combine the fractions:

The common denominator is sin^2(x). Now, let's rewrite the expression with the common denominator:

(1/sin^2(x)) - 1 = (1 - sin^2(x))/sin^2(x)

Now, recall the Pythagorean identity for sine:

sin^2(x) + cos^2(x) = 1

Rearrange it to solve for sin^2(x):

sin^2(x) = 1 - cos^2(x)

Now, substitute this value back into the expression:

(1 - sin^2(x))/sin^2(x) = (1 - (1 - cos^2(x)))/(1 - cos^2(x))

Simplify further:

(1 - 1 + cos^2(x))/(1 - cos^2(x)) = cos^2(x)/(1 - cos^2(x))

Now, apply another trigonometric identity:

cos^2(x) = 1 - sin^2(x)

Substitute this value into the expression:

cos^2(x)/(1 - cos^2(x)) = (1 - sin^2(x))/(1 - (1 - sin^2(x)))

Simplify once more:

(1 - sin^2(x))/(1 - 1 + sin^2(x)) = (1 - sin^2(x))/sin^2(x)

Lastly, recall the Pythagorean identity again:

sin^2(x) + cos^2(x) = 1

Thus, sin^2(x) = 1 - cos^2(x)

Substitute this into the expression:

(1 - (1 - cos^2(x)))/((1 - cos^2(x))) = (1 - 1 + cos^2(x))/(1 - cos^2(x)) = cos^2(x)/(1 - cos^2(x))

So, the simplified expression is: cos^2(x)/(1 - cos^2(x))