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How do you solve the equation cot²x+5 = 4 cosec x?

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Final answer:

To solve the equation cot²x + 5 = 4 cosec x, we use trigonometric identities to rewrite cot²x as cosec²x - 1, resulting in a quadratic equation in cosec x that factors into a perfect square. Taking the square root leads to finding angles for x where sine equals 1/2, such as π/6 or 5π/6 radians.

Step-by-step explanation:

To solve the equation cot²x + 5 = 4 cosec x, we need to use trigonometric identities. We know that cosec x is the reciprocal of sin x, and cot x is the reciprocal of tan x, which is also sin x/cos x. So, the equation cot²x + 5 = 4 cosec x can be rewritten using these identities.

First, express cot²x as cosec²x - 1 since cot²x = cosec²x - 1 is a Pythagorean identity. Substituting this into the original equation gives us:

(cosec²x - 1) + 5 = 4 cosec x

Then simplify the equation:

cosec²x + 4 = 4 cosec x

Now we have a quadratic equation in terms of cosec x. Since quadratic equations can be solved by bringing all the terms to one side and factoring or using the quadratic formula, we continue by subtracting 4 cosec x from both sides, giving us:

cosec²x - 4 cosec x + 4 = 0

This is a perfect square trinomial which factors to:

(cosec x - 2)² = 0

Finally, take the square root of both sides and solve for x:

cosec x - 2 = 0

cosec x = 2

sin x = 1/2

The solutions for x are the angles where the sine has a value of 1/2. Depending on the domain, these could include angles such as π/6 or 5π/6 radians (or 30° and 150°).

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