Question

Решить: 1/(1+tan^2(х))

Answer 1

Умножим числитель и знаменатель дроби на cos^2x, получим

$...=\frac{\cos^2x}{\cos^2x+\sin^2x}=\cos^2x$

P.S. $\cos^2x+\sin^2x=1$ - основное тригонометрическое тождество.

Answer 2

$\frac{1}{1 + {tg}^{2}x } = \frac{1}{1 + \frac{ {sin}^{2} x}{ {cos}^{2}x } } = \frac{1}{ \frac{ {cos}^{2} x + {sin}^{2}x }{ {cos}^{2} x} } = \frac{1}{ \frac{1}{ {cos}^{2} x} } = {cos}^{2} x$

Answer 3

To simplify the expression 1/(1+tan^2(x)), we can use a trigonometric identity. Recall the Pythagorean identity for tangent:

tan^2(x) + 1 = sec^2(x)

Now, we can rewrite the expression as:

1/(1 + tan^2(x)) = 1 / (sec^2(x))

Now, using the definition of secant:

sec(x) = 1 / cos(x)

We can rewrite the expression as:

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

Next, we invert the fraction and simplify:

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

So, the simplified expression is cos^2(x).