Question

tan²x+cot²x=2 find general solution
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Answer 1

QUESTION: Solve for x if tan²x + cot²x = 2.

SOLUTION: To solve this equation for the value of x, we need the understanding of trigonometric identities and how these identities are related. The equation above features tan (which is known as tangent) and cot (which is known as cotangent).

There is a relationship between tangent and cotangent. The cotangent of an angle is the inverse of the tangent of the same angle. Therefore, it follows mathematically that

cotx = 1/tanx

If we square both sides of this, we have

cot²x = (1/tanx)²

cot²x = 1/tan²x

This means that we can substitute 1/tan²x for cot²x in the equation. Now let's go ahead and solve.

tan²x + cot²x = 2

tan²x + 1/tan²x = 2

Now let y = tan²x, so the equation becomes

y + 1/y = 2

Multiply through by y

y² + 1 = 2y

y² – 2y + 1 = 0

Let's factorise now

y² – y – y + 1 = 0

y(y – 1) –1(y – 1) = 0

(y – 1)(y – 1) = 0

So y – 1 = 0

y = 1 twice

Remember that

y = tan²x

tan²x = y

But y = 1 so that

tan²x = 1

(tanx)² = 1

Take square root of both sides

tanx = √1

tanx = 1

x = arctan(1)

x = 45°

Wow! We have finally solved for x. The value of x is 45°.

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