Sin alpha × sin beta - cos alpha × cos beta + 1 = 0 then the value of 1 + cot alpha × tan beta is equal to

To solve the equation you've provided, let's start by simplifying the expression step by step. The equation is:

Given Equation

Sin α × Sin β - Cos α × Cos β + 1 = 0

Rearranging the Equation

We can rearrange this equation to isolate the trigonometric terms:

• Sin α × Sin β - Cos α × Cos β = -1

Using Trigonometric Identities

Now, notice that the left side resembles the cosine of the sum of two angles, specifically:

• Cos(α + β) = Cos α × Cos β - Sin α × Sin β

Thus, we can rewrite our equation using this identity:

• -(Cos(α + β)) = -1

This implies:

• Cos(α + β) = 1

Finding the Angles

The cosine of an angle equals 1 when that angle is a multiple of 2π. Therefore, we can express this as:

• α + β = 2nπ, where n is any integer.

Calculating 1 + Cot α × Tan β

Now, to find the value of 1 + Cot α × Tan β, we need to recall the definitions of cotangent and tangent:

• Cot α = 1/Tan α
• Tan β = Sin β/Cos β

So, Cot α × Tan β can be expressed as:

• Cot α × Tan β = (1/Tan α) × (Sin β/Cos β)

Combining these gives:

• Cot α × Tan β = Sin β / (Tan α × Cos β)

Using Angles' Relationship

Since we established that α + β = 2nπ, we can express β in terms of α, specifically:

• β = 2nπ - α.

This means that Sin β = Sin(2nπ - α) = Sin α and Cos β = Cos(2nπ - α) = Cos α. Using these in our expression:

• Cot α × Tan β = Sin α / (Tan α × Cos α)

Now, since Tan α = Sin α / Cos α, we can rewrite this as:

• Cot α × Tan β = Sin α / (Sin α / Cos α × Cos α) = Sin α / Sin α = 1 (assuming Sin α is not zero).

Final Calculation

Therefore, substituting back into our original expression:

• 1 + Cot α × Tan β = 1 + 1 = 2.

Conclusion

The value of 1 + Cot α × Tan β is equal to 2.