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Cot2x

Cot2x formula is an important formula in trigonometry. It is mathematically written as cot2x = (cot2x – 1)/(2cotx). Cot2x identity is also known as the double angle formula of the cotangent function in trigonometry. We can express the cot2x formula in terms of different trigonometric functions such as tan, sin, cos, and cot itself. The formula for cot2x is commonly used to find the value of the cotangent function of the double of angle x.

Further in this article, we will explore cot2x and cot^2x, and derive their formulas using trigonometric formulas and identities. We will also draw the graph cot2x trigonometric function and discuss its values at various points.

1. What is Cot2x in Trigonometry?
2. Cot2x Formula
3. Proof of Cot2x Formula
4. Cot2x Graph
5. Cot^2x (Cot Square x)
6. Cot^2x Formula
7. FAQs on Cot2x

What is Cot2x in Trigonometry?

Cot2x is an important double angle formula in trigonometry which is used to find the value of the cotangent function for double of angle x. The cot2x formula can be expressed in terms of the tangent function, sine function, cosine function, and the cotangent function itself. For example, we can write cot2x as the reciprocal of tan2x. To derive the cot2x formula, we can use the angle sum formula of the cotangent function. It is used to solve various complex trigonometric problems in maths. Let us now go through the formula of cot2x.

Cot2x Formula

Now, the formula for cot2x identity has different forms. We can write the cot2x formula as combinations of different trigonometric functions such as tan, cos, sin, and cot. For example, we can write cot2x as the ratio of cos2x and sin2x. The list of formulas of cot2x is given below:

  • cot2x = (cot^2x – 1)/(2cotx) = (cot2x – 1)/(2cotx)
  • cot2x = 1/tan2x
  • cot2x = cos2x/sin2x
  • cot2x = (1/2)(cotx – tanx)

Proof of Cot2x Formula

Now that we know the formula of cot2x, let us derive the formula using different trigonometric formulas. We have the formula of cot2x in different forms. We will derive them step-wise starting from the first one which is the main formula of cot2x given by cot2x = (cot^2x – 1)/(2cotx) = (cot2x – 1)/(2cotx).

Cot2x Formula Proof Using Cot(a+b) Formula

Since we have different forms of the formulas of cot2x, we will derive the main formula of cot2x which is cot2x = (cot^2x – 1)/(2cotx) = (cot2x – 1)/(2cotx) in this section. To derive this formula, we will use the angle sum formula of the cot. We know that we can write the angle 2x as 2x = x + x. The angle sum formula for cotangent function is cot(a+b) = (cot a cot b – 1)/(cot b + cot a). Therefore, we have

cot2x = cot(x+x)

= (cot x cot x – 1)/(cot x + cot x)

= (cot^2x – 1)/(2cot x)

= (cot2x – 1)/(2cotx)

Hence, we have derived the main formula of cot2x.

Cot2x Formula Proof In Terms of Tan, Sin and Cos

As we know that cotx and tanx are reciprocals of each other, that is, cotx = 1/tanx, therefore we can write cot2x = 1/tan2x. From here we get the second formula of cot2x. Now, since tanx can be written as the ratio of sinx and cosx, that is, tanx = sinx/cosx, using this formula, we have cot2x = 1/tan2x = 1/(sin2x/cos2x) = cos2x/sin2x. Hence, we have the third formula of the cot2x. Later, in this article, we will also derive the fourth formula of cot2x.

Cot2x Graph

Now, the image below shows the graph of cot2x which can be drawn by plotting some of its points. As we know that cotx is equal to zero when x is equal to (2n + 1)π/2, where n is an integer. Therefore, cot2x is zero when 2x = (2n+1)π/2 which implies x = (2n+1)π/4. Hence, in the graph below we can see that the graph of cot2x intersects the x-axis when x = (2n+1)π/4, where n is an integer.

Cot^2x (Cot Square x)

In this section, we will understand the cot^2x identity in trigonometry. We have different formulas and identities in trigonometry that involve cot^2x. For example, 1 + cot^2x = cosec^2x, that is, 1+cot2x = cosec2x. The cot2x formula also includes cot square x from which we can get the formula of cot^2x. In the next section, we will discuss the formulas for cot square x identity and their derivations.

Cot^2x Formula

Now, using the trigonometric formula 1 + cot^2x = cosec^2x, we can derive the cot^2x formula as cot^2x = cosec^2x – 1, that is, cot2x = cosec2x – 1. Also, using the cot2x formula cot2x = (cot^2x – 1)/(2cotx), we have cot^2x = 2cotx cot2x + 1. We can also write cot^2x as cot^2x = cos^2x/sin^2x = 1/tan^2x. Hence, the list of cot square x formula is as follows:

  • cot^2x = cosec^2x – 1
  • cot^2x = 2cotx cot2x + 1
  • cot^2x = cos^2x/sin^2x
  • cot^2x = 1/tan^2x

Important Notes on Cot2x

  • The main formula for cot2x is cot2x = (cot^2x – 1)/(2cot x).
  • The most commonly used cot^2x formula is cot^2x = cos^2x/sin^2x.
  • Using cot2x formula, the formula for cot square x is cot^2x = 2cotx cot2x + 1.

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Cot2x Examples

  1. Example 1: Establish the identity of cot2x given by cot2x = (1/2)(cotx – tanx).

    Solution: To prove the given identity for cot2x, we will use the formulas of cos2x and sin2x given by,

    • cos2x = cos2x – sin2x
    • sin2x = 2sinx cosx

    To prove cot2x = (1/2)(cotx – tanx), consider

    LHS = cot2x

    = cos2x/sin2x

    = (cos2x – sin2x)/(2sinx cosx)

    = cos2x/2sinx cosx – sin2x/2sinx cosx

    = (1/2)cosx/sinx – (1/2)sinx/cosx

    = (1/2) [cosx/sinx – sinx/cosx]

    = (1/2) (cotx – tanx) [Because cotx = cosx/sinx and tanx = sinx/cosx]

    Answer: Hence, we have established the identity cot2x = (1/2)(cotx – tanx)

  2. Example 2: Evaluate the derivative of cot2x.

    Solution: To determine the derivative of cot2x, we will use the chain rule method of differentiation. We know that the derivative of cotx is -cosec2x and the derivative of 2x is 2. Therefore, we have

    d(cot2x)/dx = d(cot2x)/d(2x) × d(2x)/dx

    = -cosec2(2x) × 2

    = -2 cosec2(2x)

    Answer: Hence, the derivative of cot2x is equal to -2 cosec2(2x).

  3. Example 3: Compute the cot2x integration.

    Solution: Now, for the integral of cot2x, we will use the formula of cot2x where it is written as cot2x = cos2x/sin2x.

    Assume sin2x = u ⇒ 2cos2x dx = du ⇒ cos2x dx = (1/2) du

    ∫cot2x dx = ∫(cos2x/sin2x) dx

    = ∫(1/2)(1/u) du

    = (1/2) ln |u| + C

    = (1/2) ln |sin2x| + C

    Answer: Hence the integral of cot2x is equal to (1/2) ln |sin2x| + C, where C is the integration constant.

Cot2x Practice Questions

FAQs on Cot2x

What is Cot2x in Trigonometry?

Cot2x is one of the important and commonly used trigonometric formulas. Mathematically, the formula for cot2x is written as cot2x = (cot^2x – 1)/(2cotx) = (cot2x – 1)/(2cotx). Cot2x can be expressed as combinations of different trigonometric functions and gives the value of cot function for double angle 2x.

What is the Formula for Cot2x?

We can express cot2x in different forms such as:

  • cot2x = (cot^2x – 1)/(2cotx) = (cot2x – 1)/(2cotx)
  • cot2x = 1/tan2x
  • cot2x = cos2x/sin2x
  • cot2x = (1/2)(cotx – tanx)

We can say that the main formula of cot2x is cot2x = (cot^2x – 1)/(2cotx) = (cot2x – 1)/(2cotx).

How to Derive Cot2x Formula?

We can derive the cot2x formula cot2x = (cot^2x – 1)/(2cotx) = (cot2x – 1)/(2cotx) using the angle sum formula of the cotangent function. Also, we can substitute 2x with x+x and then apply the formula cot(a+b) = (cot a cot b – 1)/(cot b + cot a).

What is Cot^2x Formula in Trigonometry?

The formula for cot^2x is given by

  • cot^2x = cosec^2x – 1
  • cot^2x = 2cotx cot2x + 1
  • cot^2x = cos^2x/sin^2x
  • cot^2x = 1/tan^2x

We can derive these formulas using different trigonometric formulas.

What is the Derivative of Cot2x?

The derivative of cot2x is equal to -2 cosec2(2x) which can be obtained using the chain rule of differentiation.

What is the Integral of Cot2x?

The integral of cot2x is equal to (1/2) ln |sin2x| + C, where C is the integration constant. We can evaluate this cot2x integration by substituting cot2x as cos2x/sin2x and assuming sin2x = u (an arbitrary variable).

What is the Trigonometric Identity Involving Cot^2x?

We know that the sum of cot square x and one is equal to cosec square x. Mathematically, this trigonometric identity involving cot^2x can be written as the 1 + cot^2x = cosec^2x.

What is the Derivative of Cot2x?

The derivative of cot2x is equal to -2 cosec2(2x). We can calculate the cot2x differentiation using the chain rule method of differentiation.

How to Integrate Cot2x?

The integral of cot2x is equal to (1/2) ln |sin2x| + C, where C is the integration constant. We can evaluate the integral of cot2x using the u-substitution method.

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