Tan theta over 1 + sec theta + 1 +sec theta over Tan theta= 2 csc theta
Verify the identity:
totally confusing the way you typed.
Lucky that I know the relation is supposed to say:
tanθ/(1+secθ) + (1+secθ)/tanθ = 2cscθ
LS = (tan^2 θ + (1+secθ)^2)/(tanθ(1+secθ)) , using a common denominator
= (tan^2 θ + 1 + 2secθ + sec^2 θ)((tanθ(1+secθ))
= (sec^2 θ - 1 + 1 + 2secθ + sec^2 θ)/(tanθ(1+secθ))
= 2secθ(secθ + 1)/(tanθ(1+secθ))
= 2secθ/tanθ , secθ ≠ -1
= 2(1/cosθ)(cosθ/sinθ)
= 2/sinθ
= 2cscθ
= RS
extra credit: During Reiny's proof, he came to the step where he noted
= 2secθ/tanθ , secθ ≠ -1
Are there any other values you need to watch out for ?
To solve the given equation: tan(theta)/(1 + sec(theta)) + (1 + sec(theta))/tan(theta) = 2csc(theta), we will simplify each side of the equation individually.
1. Simplifying the left-hand side (LHS) of the equation:
First, we need to simplify (1 + sec(theta)) and 1/tan(theta) separately.
- Since sec(theta) = 1/cos(theta), we can rewrite (1 + sec(theta)) as (1 + 1/cos(theta)).
- Similarly, tan(theta) = sin(theta)/cos(theta), so 1/tan(theta) = cos(theta)/sin(theta).
Now, substituting these values in the LHS of the equation:
= tan(theta)/(1 + 1/cos(theta)) + (1 + 1/cos(theta)) / (cos(theta)/sin(theta))
= tan(theta)/(cos(theta)+1)/cos(theta) + (cos(theta)+1)sin(theta)/cos(theta)
= tan(theta) * cos(theta) / (cos(theta) + 1) + sin(theta) + sin(theta)*cos(theta) / (cos(theta) + 1)
2. Simplifying the right-hand side (RHS) of the equation:
Here, we need to work on 2csc(theta).
- Since csc(theta) = 1/sin(theta), we can rewrite 2csc(theta) as 2/sin(theta).
Now, we have:
LHS = tan(theta) * cos(theta) / (cos(theta) + 1) + sin(theta) + sin(theta)*cos(theta) / (cos(theta) + 1)
RHS = 2/sin(theta)
3. Combining the LHS and RHS:
Now, we can equate the LHS and RHS of the equation, i.e., LHS = RHS, and solve for theta.
tan(theta) * cos(theta) / (cos(theta) + 1) + sin(theta) + sin(theta)*cos(theta) / (cos(theta) + 1) = 2/sin(theta)
To further simplify and solve this equation, we will multiply both sides by sin(theta) * (cos(theta) + 1) to eliminate the denominators. Then, we can proceed with solving for theta.