Problem 47 - Precalculus Essentials

Question

Find the exact value of each expression. Do not use a calculator. $$\csc 37^{\circ} \sec 53^{\circ}-\tan 53^{\circ} \cot 37^{\circ}$$

Step-by-Step Solution

Answer

The expression simplifies to 0.

Step 1: Understanding the given expression

The expression given is \( \csc 37^{\circ} \sec 53^{\circ} -\tan 53^{\circ} \cot 37^{\circ} \) where \(\csc, \sec, \tan, \cot\) are cosecant, secant, tangent and cotangent functions respectively.

Step 2: Simplifying the expression using identities

The given terms can be expressed in terms of more familiar sine, cosine, and tangent. The cosecant of an angle is the reciprocal of the sine of an angle and the secant of an angle is the reciprocal of the cosine. So, the first term can be rewritten as \( \frac{1}{\sin 37^{\circ}} \cdot \frac{1}{\cos 53^{\circ}} \). Similarly, the tangent of an angle is the sine of the angle divided by the cosine, and the cotangent is the reciprocal of the tangent. So, the second term can be rewritten as \( \frac{\sin 53^{\circ}}{\cos 53^{\circ}} \cdot \frac{\cos 37^{\circ}}{\sin 37^{\circ}} \).

Step 3: Simplifying further

Notice that \( 37^{\circ} \) and \( 53^{\circ} \) are complementary angles, since they add up to \( 90^{\circ} \). For the sine and cosine functions, the sine of an angle equals the cosine of its complementary angle, that is, \( \sin 37^{\circ} = \cos 53^{\circ} \) and \( \cos 37^{\circ} = \sin 53^{\circ} \). Substituting into the expression, both terms simplify to 1, and you are left with \( 1 - 1 \).

Step 4: Final simplification

On subtracting the two terms, the result is \( 0 \).