1/tan beta +tan beta=sec^2 beta/tan beta
1/tanβ + tanβ
(1+tan^2 β)/tanβ
sec^2 β/tan
since sec^2 β = 1+tan^2 β
To solve this equation, let's simplify both sides individually and then compare them.
Starting with the left side of the equation:
1/tan(beta) + tan(beta)
To simplify this expression, we need to find a common denominator. The denominator of the first term is just tan(beta), while the second term has a denominator of 1. We can multiply the first term by 1 in the form of tan(beta)/tan(beta). This gives us:
1/tan(beta) + tan(beta) * tan(beta) / tan(beta)
Simplifying this further gives:
1/tan(beta) + tan^2(beta) / tan(beta)
Now, we can combine these two terms by finding a common denominator, which is just tan(beta). This gives us:
(1 + tan^2(beta)) / tan(beta)
Now let's simplify the right side of the equation:
sec^2(beta) / tan(beta)
The secant squared (sec^2) of beta is equal to (1/cos(beta))^2, which means that sec^2(beta) can be rewritten as 1/cos^2(beta). Substituting this into the equation gives us:
(1/cos^2(beta)) / tan(beta)
To divide by a fraction, we can multiply by its reciprocal. In this case, we can multiply by cos^2(beta) / 1:
(1/cos^2(beta)) / tan(beta) * (cos^2(beta) / 1)
Simplifying this expression gives us:
cos^2(beta) / (cos^2(beta) * tan(beta))
Now, we can simplify the expression further by canceling out the common factor of cos^2(beta):
1 / tan(beta)
Now we have simplified both sides of the equation and they are equal:
(1 + tan^2(beta)) / tan(beta) = 1 / tan(beta)
To solve for beta, we can cross-multiply:
1 + tan^2(beta) = 1
Next, subtract 1 from both sides of the equation:
tan^2(beta) = 0
Taking the square root of both sides gives us:
tan(beta) = 0
Now, to find the values of beta that satisfy this equation, we need to find the values where the tangent function is equal to zero. This occurs at multiples of pi radians or 180 degrees (beta = n*pi), where n is an integer.