Find the exact value of the expression. cos [ tan^-1 (4/3) - cos^-1 (7/25) ]

Question

SnapAndSolve AI · Accepted Answer

To find the exact value of the expression, we need to evaluate the expression inside the cosine function:

Let θ be such that tan(θ) = 4/3. Then, we can use the identity tan(θ) = opposite/adjacent to form a right triangle with opposite side 4 and adjacent side 3. The hypotenuse is √(4² + 3²) = √25 = 5. Therefore, cos(θ) = adjacent/hypotenuse = 3/5.
Let φ be such that cos(φ) = 7/25.
We need to find cos(θ - φ) using the identity cos(θ - φ) = cos(θ)cos(φ) + sin(θ)sin(φ).
We already have cos(θ) = 3/5 and cos(φ) = 7/25.
To find sin(θ), use sin²(θ) = 1 - cos²(θ): sin(θ) = √(1 - (3/5)²) = 4/5.
To find sin(φ), use sin²(φ) = 1 - cos²(φ): sin(φ) = √(1 - (7/25)²) = 24/25.
Now, substitute these values into the identity: cos(θ - φ) = (3/5)(7/25) + (4/5)(24/25).
Simplify: cos(θ - φ) = 21/125 + 96/125 = 117/125.

Therefore, the exact value of the expression is 117/125.

Answer