We have to solve for the problem below to make sure that it is equivalent to 2csc(x). However, I keep getting 2/2sin(x) and not 2/sin(x), which is what 2csc(x) is.
tan(x)/(1+sec(x))+ (1+sec(x))/tan(x)= 2csc(x)
tan(x)/(1+sec(x))+ (1+sec(x))/tan(x)
= tan x/[(cosx+1)/cosx]
+ [(cosx+1)/cosx]/tanx
= sin x/(cosx+1) + (cosx +1)/sinx
= [sin^2 x + (cosx +1)^2]/[sinx(1 + cosx)]
= (sin^2x + cos^2x + 2 cos x + 1)/[sinx(1 + cosx)]
= 2 (1 + cosx)/[sinx(1 + cosx)]
= 2/sin x = 2 csc x
To solve the given equation and show that it is equivalent to 2csc(x), we can follow these steps:
Step 1: Simplify the left side of the equation.
Start by multiplying the numerator and denominator of the first fraction, tan(x), by sec(x). Similarly, multiply the numerator and denominator of the second fraction, 1+sec(x), by tan(x).
Now, the equation becomes:
(tan(x) * sec(x))/(sec(x) + 1) + (1 + sec(x)) * (tan(x)/sec(x))
Step 2: Simplify further.
Apply the distributive property to both fractions:
(tan(x) * sec(x))/(sec(x) + 1) + tan(x) + tan(x) * sec(x)
Next, rearrange the terms in the numerator of the first fraction, tan(x) * sec(x), to sec(x) * tan(x) (since multiplication is commutative).
(sec(x) * tan(x))/(sec(x) + 1) + tan(x) + tan(x) * sec(x)
Step 3: Find a common denominator.
To add the fractions, we need to find a common denominator. The common denominator here is (sec(x) + 1).
Rewrite all the terms with the common denominator:
(sec(x) * tan(x))/(sec(x) + 1) + ((tan(x) * (sec(x) + 1))/(sec(x) + 1)) + (tan(x) * sec(x))/(sec(x) + 1)
Step 4: Combine the fractions.
Combine the fractions into a single fraction:
[sec(x) * tan(x) + tan(x) * (sec(x) + 1) + tan(x) * sec(x)] / (sec(x) + 1)
Step 5: Simplify the numerator.
Apply the distributive property to the terms in the numerator:
[sec(x) * tan(x) + sec(x) * tan(x) + tan(x)] / (sec(x) + 1)
Combine like terms:
[2 sec(x) * tan(x) + tan(x)] / (sec(x) + 1)
Step 6: Simplify further.
Divide both the numerator and denominator by cos(x) (since sec(x) = 1/cos(x)):
[2 tan(x) + tan(x)] / (1/cos(x) + 1)
Combine like terms in the numerator:
[3 tan(x)] / (1/cos(x) + 1)
Step 7: Simplify the denominator.
To add the fraction in the denominator, we need to find a common denominator. The common denominator is cos(x). Rewrite the fraction:
[3 tan(x)] / [(1 + cos(x))/cos(x)]
Step 8: Simplify further.
Divide the numerator by cos(x) and simplify the denominator:
3 tan(x) / (1/cos(x) + cos(x)/cos(x))
Simplify the denominator by adding the fractions in the denominator:
3 tan(x) / [(1 + cos(x))/cos(x)]
Step 9: Simplify the numerator.
Multiply the numerator by the reciprocal of the denominator:
3 tan(x) * (cos(x)/(1 + cos(x)))
Step 10: Simplify further.
Apply the distributive property to multiply the terms:
(3 tan(x) * cos(x)) / (1 + cos(x))
Use the identity sin(x) = tan(x)/cos(x) to rewrite the numerator:
(3 sin(x)) / (1 + cos(x))
Now, we have shown that the given equation, tan(x)/(1+sec(x)) + (1+sec(x))/tan(x) is equivalent to 2csc(x) which is equal to (3 sin(x)) / (1 + cos(x)).