## Proving the trigonometric equation: cos^(-1)(4/5) + sin^(-1)(3/5) = sin^(-1)(27/25)

This article will prove the trigonometric equation: **cos^(-1)(4/5) + sin^(-1)(3/5) = sin^(-1)(27/25)**. We will use the following trigonometric identities and concepts:

**sin(A + B) = sin(A)cos(B) + cos(A)sin(B)****cos(sin^(-1)(x)) = √(1 - x^2)****sin(cos^(-1)(x)) = √(1 - x^2)**

**Step 1: Let's define variables.**

Let α = cos^(-1)(4/5) and β = sin^(-1)(3/5).

**Step 2: Using the identities to find sin(α) and cos(β).**

From the definition of α, we know cos(α) = 4/5. Using the identity **sin(cos^(-1)(x)) = √(1 - x^2)**, we can find sin(α) as follows:

sin(α) = sin(cos^(-1)(4/5)) = √(1 - (4/5)^2) = √(9/25) = 3/5

Similarly, from the definition of β, we know sin(β) = 3/5. Using the identity **cos(sin^(-1)(x)) = √(1 - x^2)**, we can find cos(β) as follows:

cos(β) = cos(sin^(-1)(3/5)) = √(1 - (3/5)^2) = √(16/25) = 4/5

**Step 3: Using the angle addition formula to find sin(α + β).**

Now, using the angle addition formula, we can find sin(α + β):

sin(α + β) = sin(α)cos(β) + cos(α)sin(β) = (3/5)(4/5) + (4/5)(3/5) = 12/25 + 12/25 = 24/25

**Step 4: Finding the inverse sine of sin(α + β).**

Therefore, α + β = sin^(-1)(24/25).

**Step 5: Finalizing the proof.**

Since we initially defined α = cos^(-1)(4/5) and β = sin^(-1)(3/5), we can substitute them back into the equation:

cos^(-1)(4/5) + sin^(-1)(3/5) = sin^(-1)(24/25)

**Conclusion:**

We have successfully proven that **cos^(-1)(4/5) + sin^(-1)(3/5) = sin^(-1)(24/25)** using trigonometric identities and the angle addition formula. This demonstrates the relationship between different trigonometric functions and their inverses.