Verify the Identity:
sec(x/2)=the square root of (2 tanx/(tanx sinx))
cannot be verified since it is not true.
All we need is one example where it does NOT work
e.g. let x = 60
LS = sec (30) = 1.1547
RS = √(2(1.7321)/((1.7321)(.5))
= √4 = 2
LS ≠ RS, so no identity
To verify the identity, we'll start by manipulating the right-hand side of the equation to match the left-hand side:
Given: sec(x/2) = √(2 tan(x) / (tan(x) sin(x)))
Recall the following trigonometric identities:
1. sec(x) = 1 / cos(x)
2. tan(x) = sin(x) / cos(x)
Let's simplify the right-hand side:
√(2 tan(x) / (tan(x) sin(x)))
= √(2 (sin(x) / cos(x)) / ((sin(x) / cos(x)) sin(x)))
= √(2 sin(x) / (sin(x)² / cos(x)))
= √(2 sin(x) cos(x) / sin(x)²)
= √(2 cos(x) / sin(x))
Now, let's simplify the left-hand side:
sec(x/2) = 1 / cos(x/2)
To match the right-hand side, we need to manipulate this equation:
1 / cos(x/2)
= 1 / √(1 - sin²(x/2)) || using the identity cos(x/2) = √(1 - sin²(x/2))
= 1 / √(1 - (1 - cos(x)) / 2) || substituting sin²(x/2) = (1 - cos(x)) / 2
= 1 / √((2 - (1 - cos(x))) / 2)
= 1 / √((1 + cos(x)) / 2)
= √(2 / (1 + cos(x)))
Now, we have:
√(2 / (1 + cos(x))) = √(2 cos(x) / sin(x))
To verify this identity, we can square both sides:
(2 / (1 + cos(x))) = (2 cos(x) / sin(x))
Multiplying both sides by (1 + cos(x)) * sin(x):
2 sin(x) = 2 cos(x) * (1 + cos(x))
Expanding:
2 sin(x) = 2 cos(x) + 2 cos²(x)
Dividing both sides by 2:
sin(x) = cos(x) + cos²(x)
Using the identity sin²(x) + cos²(x) = 1, we can substitute sin²(x) = 1 - cos²(x):
1 - cos²(x) = cos(x) + cos²(x)
Rearranging:
2 cos²(x) + cos(x) - 1 = 0
Now, let's solve this quadratic equation. Factoring or using the quadratic formula will give us:
(cos(x) - 1)(2cos(x) + 1) = 0
From this, we get two possible solutions:
cos(x) - 1 = 0 or 2cos(x) + 1 = 0
cos(x) = 1 or cos(x) = -1/2
The solutions for cos(x) = 1 are x = 2πn, where n is an integer.
The solutions for cos(x) = -1/2 are x = 2π/3 + 2πn or x = 4π/3 + 2πn, where n is an integer.
Therefore, we have verified that sec(x/2) = √(2 tan(x) / (tan(x) sin(x))].
To verify the given identity, we need to simplify the left side of the equation and compare it to the right side of the equation. Let's break it down step by step:
1. Start with the left side of the equation: sec(x/2).
Recall the identity: sec(x) = 1/cos(x). We can use this to rewrite sec(x/2) as 1/cos(x/2).
2. Simplify the right side of the equation: the square root of (2 tanx/(tanx sinx)).
Notice that the square root simplifies to the power of 1/2. Therefore, the right side becomes: (2 tanx/(tanx sinx))^(1/2).
3. Now, let's simplify the right side further:
We can rewrite (2 tanx/(tanx sinx))^(1/2) as √(2 tanx)/(√(tanx sinx)).
Taking the square root of a fraction means taking the square root of the numerator and the square root of the denominator separately:
√(2 tanx) = √2 * √(tanx), and √(tanx sinx) = √tanx * √sinx.
4. As a result, the right side becomes: (√2 * √(tanx)) / (√tanx * √sinx).
5. To simplify, let's combine the fractions in the numerator and denominator:
(√2 * √(tanx)) / (√tanx * √sinx) = (√2/√tanx) * (√(tanx)/√sinx).
6. Observe that (√(tanx)/√tanx) cancels out, leaving us with the simplified right side of the equation as: √2/√sinx.
7. Compare the left and right sides of the equation:
Left side: 1/cos(x/2)
Right side: √2/√sinx
8. To simplify further, we need to express both sides of the equation using the same trigonometric function. We can do this by using the identity: cos(x) = sin(π/2 - x).
Substitute x/2 with π/2 - x in the left side of the equation:
1/cos(x/2) = 1/cos(π/2 - x).
9. Simplify the expression further using the identity: 1/cos(x) = sec(x).
Therefore, 1/cos(π/2 - x) = sec(π/2 - x).
10. Compare the left side (sec(π/2 - x)) and the right side (√2/√sinx) of the equation.
Thus, by simplifying both sides of the equation and comparing them, we have verified the given identity: sec(x/2) = √2/√sinx.