Solve :

sec^2(x)= �ã3*tanx + 1 on [0,radian-180 degrees) - couldn't find the symbol for radian on computer

please try and explain how you got your answer best as yu can thanks so much :)

To solve the equation sec^2(x) = 3tan(x) + 1, we will follow these steps:

Step 1: Rewrite the equation using trigonometric identities.
We know that sec^2(x) = 1/cos^2(x) and tan(x) = sin(x)/cos(x). Substituting these in the equation, we have:

1/cos^2(x) = 3sin(x)/cos(x) + 1

Step 2: Multiply both sides of the equation by cos^2(x) to eliminate the denominators.
This results in:

1 = 3sin(x)cos(x) + cos^2(x)

Step 3: Rearrange the equation to make it quadratic in terms of cos(x).
Rearranging the terms, we get:

0 = cos^2(x) + 3sin(x)cos(x) + 1 - 1

0 = cos^2(x) + 3sin(x)cos(x)

Step 4: Factor the quadratic equation.
To factor the equation, notice that the left side resembles a product of two terms involving cos(x). We can rewrite the equation as:

0 = cos(x)(cos(x) + 3sin(x))

Step 5: Solve each factor separately.
Setting cos(x) = 0, we find one solution: x = π/2 + nπ, where n is an integer.

Next, setting cos(x) + 3sin(x) = 0, we solve for x. This equation does not directly factor, so we will use the sine and cosine ratio identity, sin(x) = √(1-cos^2(x)).

cos(x) + 3√(1-cos^2(x)) = 0
3√(1-cos^2(x)) = -cos(x)
9(1-cos^2(x)) = cos^2(x)
9 - 9cos^2(x) = cos^2(x)
10cos^2(x) - 9 = 0

Using the quadratic formula, we find cos(x) = ±√(9/10).

Taking the inverse cosine, we have two solutions: x = ±arccos(√(9/10)).

Step 6: Find the values of x in the given interval.
The interval [0, radian-180 degrees) represents a range of values from 0 radians to slightly before π radians (180 degrees) counterclockwise.

For the first solution x = π/2 + nπ, we need to find the n values that make x fall within the given range. In this case, n = 0 gives us x = π/2, which is within the given interval.

For the two solutions x = ±arccos(√(9/10)), we need to determine if they fall within the given range. When we evaluate the inverse cosine, we find that arccos(√(9/10)) is approximately 0.4845 radians (27.77 degrees).

Therefore, one solution is x ≈ 0.4845 radians (27.77 degrees), which is within the given interval.

Step 7: Summarize the solutions.
The solutions for x in the given interval are:

x = π/2 + nπ, where n is an integer
x ≈ 0.4845 radians (27.77 degrees)

To solve the equation sec^2(x) = √3*tan(x) + 1 on the interval [0, π-180 degrees), we'll follow these steps:

Step 1: Rewrite the Equation
Since the equation involves trigonometric functions, we'll use trigonometric identities to simplify it. Recall the following:

- sec^2(x) = 1 + tan^2(x)
- tan(x) = sin(x)/cos(x)

By substituting these identities into the original equation, we get:
1 + tan^2(x) = √3*sin(x)/cos(x) + 1

Step 2: Combine Terms
Now, let's combine like terms and get rid of denominators:
cos^2(x) + sin^2(x) = √3*sin(x)*cos(x) + cos^2(x)

Step 3: Simplify
Cancel out the cos^2(x) terms:
sin^2(x) = √3*sin(x)*cos(x)

Step 4: Solve
Now, solve for sin(x) = 0 and sin(x) ≠ 0:

a) sin(x) = 0
If sin(x) = 0, it means that x is an integer multiple of π.
So, x = 0, π, 2π, ...

b) sin(x) ≠ 0
Divide both sides by sin(x):
sin(x) = √3*cos(x)

Divide both sides by cos(x):
tan(x) = √3

Now, find the angle whose tangent (tan) is equal to √3. In the interval [0, π-180 degrees), this angle is π/3.

Therefore, x = π/3.

Summary:
The solutions to the equation sec^2(x) = √3*tan(x) + 1 on the interval [0, π-180 degrees) are x = 0, π, 2π, ... and x = π/3.