Question

1 / cos²x = 2tg²x....

Answer 1

Зная, что $\mathrm{tg^2x=\dfrac{1}{\cos^2x} -1}$, имеем

$\mathrm{\dfrac{1}{\cos^2x}=2\cdot \bigg(\dfrac{1}{\cos^2x}-1\bigg)}\\ \\ \mathrm{\dfrac{1}{\cos^2x}=2}\\\\ \mathrm{\cos^2x=\frac{1}{2}} ~~~\Rightarrow~~~\mathrm{\dfrac{1+\cos2x}{2}=\dfrac{1}{2}} \\ \\ \mathrm{\cos 2x=0}~~~\Rightarrow~~ \boxed{\mathrm{x=\frac{\pi}{4}+\frac{\pi n}{2},n\in \mathbb{Z}}}$

Answer 2

решение на фото внизу

Answer 3

To solve the trigonometric equation 1/cos^2(x) = 2tan^2(x), we'll need to manipulate the expressions using trigonometric identities. Let's start by expressing tan^2(x) in terms of sin(x) and cos(x) using the identity:

tan^2(x) = sin^2(x) / cos^2(x)

Now, the equation becomes:

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

Next, let's simplify the right-hand side by canceling out the common factor of cos^2(x) in the numerator and denominator:

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

Now, we want everything on one side of the equation. Let's move the term with 1/cos^2(x) to the right-hand side:

1 / cos^2(x) - 2 * sin^2(x) = 0

To combine the fractions, we need a common denominator. The common denominator is cos^2(x):

(1 - 2 * cos^2(x) * sin^2(x)) / cos^2(x) = 0

Now, we'll use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to replace sin^2(x) with (1 - cos^2(x)):

(1 - 2 * cos^2(x) * (1 - cos^2(x))) / cos^2(x) = 0

Now, let's simplify the expression:

(1 - 2 * (cos^2(x) - cos^4(x))) / cos^2(x) = 0

(1 - 2cos^2(x) + 2cos^4(x)) / cos^2(x) = 0

Now, we can solve for cos^2(x) by multiplying both sides by cos^2(x):

1 - 2cos^2(x) + 2cos^4(x) = 0

This equation is now in terms of cos^2(x). It's a quadratic equation in cos^2(x). Let's solve for cos^2(x):

2cos^4(x) - 2cos^2(x) + 1 = 0

Now, we can factor the quadratic equation:

(2cos^2(x) - 1)^2 = 0

Now, take the square root of both sides:

2cos^2(x) - 1 = 0

Now, solve for cos^2(x):

2cos^2(x) = 1

cos^2(x) = 1/2

Finally, take the square root of both sides to find the possible values of cos(x):

cos(x) = ±√(1/2)

cos(x) = ±(√2 / 2)

So the solutions for cos(x) are:

1. cos(x) = √2 / 2
2. cos(x) = -√2 / 2

Now, to find the values of x, we'll consider the unit circle or use a calculator to find the angles where the cosine is equal to these values. The solutions for x are:

1. x = π/4 + 2nπ, where n is an integer.
2. x = 3π/4 + 2nπ, where n is an integer.

These are the solutions to the trigonometric equation 1 / cos^2(x) = 2tan^2(x).