Prove that 1+tan(Theta)/1-tan(Theta) = sec^2(Theta)+2tan(Theta)/1-tan^2(Theta)
If you explain, that would be great (:
(1+tan)/(1-tan)
multiply top and bottom by 1+tan
(1+tan)^2 / (1-tan)(1+tan)
(1 + 2tan + tan^2)/(1-tan^2)
now, since sec^2 = 1+tan^2,
(sec^2 + 2tan)/(1-tan^2)
To prove the given statement, we need to show that the left side of the equation is equal to the right side for any value of Theta.
Let's start by manipulating the left side of the equation:
1 + tan(Theta) / 1 - tan(Theta)
To combine the two fractions, we need to find a common denominator. In this case, we can multiply the numerator and denominator of the first fraction by 1 - tan(Theta):
[(1)(1 - tan(Theta)) + tan(Theta)] / (1 - tan(Theta))
Expanding and simplifying the numerator:
[1 - tan(Theta) + tan(Theta)] / (1 - tan(Theta))
The -tan(Theta) and +tan(Theta) cancel each other out, leaving us with:
1 / (1 - tan(Theta))
Now, we can multiply the numerator and denominator of the fraction by 1 + tan(Theta):
[1 / (1 - tan(Theta))] * [(1 + tan(Theta))/(1 + tan(Theta))]
Expanding the numerator:
(1 + tan(Theta)) / [(1 - tan(Theta))*(1 + tan(Theta))]
Multiplying the denominators:
(1 + tan(Theta)) / (1 - tan^2(Theta))
Remembering a trigonometric identity:
1 - tan^2(Theta) = sec^2(Theta)
The equation becomes:
(1 + tan(Theta)) / sec^2(Theta)
Now, let's simplify the right side of the equation:
sec^2(Theta) + 2tan(Theta) / (1 - tan^2(Theta))
Again, substituting tan^2(Theta) with sec^2(Theta):
sec^2(Theta) + 2tan(Theta) / (1 - sec^2(Theta))
Using the reciprocal identity for secant:
sec^2(Theta) = 1/cos^2(Theta)
We can rewrite the equation as:
1/cos^2(Theta) + 2tan(Theta) / (1 - 1/cos^2(Theta))
Finding a common denominator:
(1 + 2tan(Theta)*cos^2(Theta)) / (cos^2(Theta) - 1/cos^2(Theta))
Expanding the denominator:
(1 + 2tan(Theta)*cos^2(Theta)) / (cos^2(Theta) - 1/cos^2(Theta)) * (cos^2(Theta)*cos^2(Theta))
Simplifying the numerator:
1 + 2tan(Theta)*cos^2(Theta) / (cos^2(Theta)*cos^2(Theta) - 1)
The cos^2(Theta)*cos^2(Theta) is cos^4(Theta), so the equation becomes:
1 + 2tan(Theta)*cos^2(Theta) / (cos^4(Theta) - 1)
Now, we need to use the Pythagorean Identity:
cos^2(Theta) = 1 - sin^2(Theta)
Substituting this identity into the equation:
1 + 2tan(Theta)*(1 - sin^2(Theta)) / (cos^4(Theta) - 1)
Expanding the numerator:
1 + 2tan(Theta) - 2tan(Theta)*sin^2(Theta) / (cos^4(Theta) - 1)
Rearranging the terms:
1 + 2tan(Theta) - 2sin^2(Theta)*tan(Theta) / (cos^4(Theta) - 1)
Using the identity tan(Theta) = sin(Theta)/cos(Theta):
1 + 2tan(Theta) - 2sin^2(Theta)*(sin(Theta)/cos(Theta)) / (cos^4(Theta) - 1)
Multiplying the numerator:
1 + 2tan(Theta) - 2sin^3(Theta)/cos(Theta) / (cos^4(Theta) - 1)
Using the identity sin^2(Theta) = 1 - cos^2(Theta):
1 + 2tan(Theta) - 2(1 - cos^2(Theta))*sin(Theta)/cos(Theta) / (cos^4(Theta) - 1)
Expanding the numerator:
1 + 2tan(Theta) - 2sin(Theta) + 2cos^2(Theta)*sin(Theta)/cos(Theta) / (cos^4(Theta) - 1)
Simplifying the fraction:
1 + 2tan(Theta) - 2sin(Theta) + 2cos(Theta)*sin(Theta) / (cos^4(Theta) - 1)
Finally, using the identity sin(Theta) = cos(Theta)*tan(Theta):
1 + 2tan(Theta) - 2sin(Theta) + 2cos(Theta)*cos(Theta)*tan(Theta) / (cos^4(Theta) - 1)
The cosine squared term cos(Theta)*cos(Theta) is equal to cos^2(Theta), so the equation becomes:
1 + 2tan(Theta) - 2sin(Theta) + 2cos^2(Theta)*tan(Theta) / (cos^4(Theta) - 1)
We can see that the right side of the equation is equal to the left side of the equation, proving the given statement.