Without a calculator, find the exact value of csc20° − cot 40° , showing work.

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csc20° − cot 40°

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What is csc(20) - cot(40)? or csc(pi/9) - cot(2pi/9)

You will see solution step by step.

To find the exact value of csc20° − cot 40° without a calculator, we need to use some trigonometric identities and basic trigonometry principles.

First, let's start by finding the value of csc20°.

The reciprocal of sine is the cosecant function, so we can find csc20° by taking the reciprocal of sin20°. The exact value of sin20° is not easily calculable, so we can use the half-angle identity for sine to simplify it.

The half-angle identity for sine is:
sin(x/2) = ±√[(1 - cos(x)) / 2].

In this case, let's use the positive square root, so we have:
sin(20°/2) = √[(1 - cos20°) / 2].

Now, let's solve for cos20°. We can use the double-angle identity for cosine to simplify it:
cos(2x) = 2cos^2(x) - 1.

In this case, x is 20°. So, substituting x = 20° into the double-angle identity, we have:
cos(40°) = 2cos^2(20°) - 1.

Now, let's solve for cos20° using a quadratic equation. Rearranging the equation, we have:
2cos^2(20°) = cos(40°) + 1.

Dividing both sides by 2, we get:
cos^2(20°) = (cos(40°) + 1) / 2.

Now, taking the square root of both sides, we get:
cos(20°) = ±√[(cos(40°) + 1) / 2].

Again, let's use the positive square root, so we have:
cos(20°) = √[(cos(40°) + 1) / 2].

Now, substituting this value of cos(20°) back into our expression for csc20°, we get:
csc20° = 1 / sin(20°) = 1 / √[(1 - cos20°) / 2].

Since we already found the expression for cos(20°), we can substitute it into the expression for csc20°:
csc20° = 1 / √[(1 - √[(cos(40°) + 1) / 2]) / 2].

Next, let's find the value of cot40°.

Cotangent (cot) is the reciprocal of tangent (tan), so we can find cot40° by taking the reciprocal of tan40°. The exact value of tan40° is not easily calculable, so we can use the cotangent identity to simplify it.

The cotangent identity is:
cot(x) = 1 / tan(x).

In this case, cot40° = 1 / tan40°.

Now, let's substitute the values we found for csc20° and cot40° into our expression:
csc20° - cot40° = 1 / √[(1 - √[(cos(40°) + 1) / 2]) / 2] - 1 / tan40°.

Since we already found the expression for cos(40°), we can substitute it into our expression:
csc20° - cot40° = 1 / √[(1 - √[(√[(cos(40°) + 1) / 2]) + 1) / 2]) - 1 / tan40°.

At this point, we have simplified the expressions as much as possible without a calculator. However, the final exact value involves square roots and trigonometric functions that are difficult to evaluate without a calculator.