Solve following equations

¡¼tan¡½^(2 ) x-5tanx=-6

To solve the equation tan^2(x) - 5tan(x) = -6, we can use the substitution method. Let's substitute tan(x) with another variable, say u.

Let u = tan(x).

Now, the equation becomes u^2 - 5u = -6.

Rearranging the equation, we have u^2 - 5u + 6 = 0.

We can now factorize the equation to find the values of u.

(u - 2)(u - 3) = 0.

Setting each factor equal to zero gives us:

u - 2 = 0 or u - 3 = 0.

Solving these equations, we find two solutions:
u = 2 or u = 3.

However, we need to remember that u = tan(x).

So, we have two possible solutions for x:

tan(x) = 2 or tan(x) = 3.

To find the values of x, we can use inverse trigonometric functions such as arctan.

Taking the arctan of both sides of the equations, we get:

x = arctan(2) or x = arctan(3).

These are the solutions to the given equation.