Trig problem help?

Question

Its about trig subsitution. the problem is square root of 4 + x(squared)
x= 2tan(of theta)
i subsituted x into the equation but i dont knoew how to get the next answer. the book says ists square root of 4(1 +tan(squared of theta) )
how do you get to that?

Answer 1

You would get this with the substitution (I am using x for theta):

4 + (2tanx)^2
= 4 + 4(tanx)^2
= 4(1 + (tanx)^2)
= 4 (sec x)^2
then take the squre root to get 2|sec x|

Take another look at your 3 Pythagorean identities:

cos^2 x + sin^2 x = 1
1 + tan^2 x = sec^2 x
cot^2 x + 1 = csc^2 x

Many times you can make an appropriate substitution to turn a binomial into a monomial as occurred above. I have seen precalc books do this, but the actual process is used in calculus as an integration technique.

Answer 2

x = 2 tan θ, so x^2 = 4 tan^2 θ.
So √(4 + x^2) = √(4 + 4 tan^2 θ)
= √(4(1 + tan^2 θ))
(this is just taking out the common factor of 4).
You'd probably continue:
= √4 √(sec^2 θ)
= 2 |sec θ|.

Answer 3

2/(a million+cosx) - tan²(x/2) = a million Taking LHS 2/(a million+cosx) - (sin²x/2 / cos²x/2) 2/(2cos²x/2-a million +a million) -(sin²x/2 / cos²x/2) a million/cos²x/2 -(sin²x/2/cos²x/2) a million-sin²x/2 / cos²x/2 cos²x/2 / cos²x/2 a million (Proved ) Trignometric Identities Used :- cos2x = 2cos²x -a million sin²x + cos²x = a million Tanx = sinx / cosx