Evaluate each expression without using a calculator.
cos (tan⁻¹ (5/12) - tan⁻¹ (3/4))

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Recognize that the expression involves the cosine of a difference of two inverse tangent values: \(\cos\left(\tan^{-1}\left(\frac{5}{12}\right) - \tan^{-1}\left(\frac{3}{4}\right)\right)\). We can use the cosine difference identity to simplify this.

Recall the cosine difference identity: \(\cos(A - B) = \cos A \cos B + \sin A \sin B\). Here, let \(A = \tan^{-1}\left(\frac{5}{12}\right)\) and \(B = \tan^{-1}\left(\frac{3}{4}\right)\).

Find \(\cos A\) and \(\sin A\) by considering a right triangle where the opposite side is 5 and adjacent side is 12 (since \(\tan A = \frac{5}{12}\)). Use the Pythagorean theorem to find the hypotenuse: \(\sqrt{5^2 + 12^2}\).

Similarly, find \(\cos B\) and \(\sin B\) by considering a right triangle where the opposite side is 3 and adjacent side is 4 (since \(\tan B = \frac{3}{4}\)). Use the Pythagorean theorem to find the hypotenuse: \(\sqrt{3^2 + 4^2}\).

Substitute the values of \(\cos A\), \(\sin A\), \(\cos B\), and \(\sin B\) into the identity \(\cos(A - B) = \cos A \cos B + \sin A \sin B\) and simplify the expression step-by-step.

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Inverse Tangent Function (tan⁻¹)

The inverse tangent function, tan⁻¹(x), gives the angle whose tangent is x. It is used to find an angle when the ratio of the opposite side to the adjacent side in a right triangle is known. Understanding this helps convert the given ratios into angles for further trigonometric manipulation.

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Inverse Tangent

Cosine of a Difference of Angles Formula

The formula cos(A - B) = cos A cos B + sin A sin B allows evaluation of the cosine of the difference between two angles. This identity is essential to break down the expression cos(tan⁻¹(5/12) - tan⁻¹(3/4)) into simpler trigonometric terms involving sine and cosine of each angle.

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Verifying Identities with Sum and Difference Formulas

Right Triangle Trigonometric Ratios

Using the given tangent values, one can construct right triangles to find the sine and cosine of the angles. For example, tan⁻¹(5/12) corresponds to a triangle with opposite side 5 and adjacent side 12, allowing calculation of hypotenuse and thus sine and cosine values needed for the formula.

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