Question

In the given figure find tan P - cot R

Answer 1

By applying the Pythagoras theorem to \(ΔPQR,\) we get \( (PR)^ 2 = (PQ)^ 2 +( QR)^ 2 \). Substituting the given values, we have \( (13 \ cm) ^2 = (12\ cm)^ 2 + (QR)^ 2 \), which simplifies to \( 169 \ cm^2 = 144 \ cm^2 + (QR)^ 2 \). This further simplifies to \( 25\ cm^2 = (QR)^ 2 \), thus \( QR = 5 \ cm\).

The tangent of angle P is defined as \( \text{ tan P} = \frac{\text{Side}\ \text{ Opposite}\ \text{ to}\ ∠P }{\text{Side}\ \text{ Adjacent}\ \text{ to}\ ∠P}\), which is \( =\frac{QR}{PQ}=\frac{5}{12}\). The cotangent of angle R is defined as \( \text{ cot R} = \frac{\text{Side}\ \text{Adjacent}\ \text{ to}\ ∠R }{\text{Side}\ \text{ Opposite}\ \text{ to}\ ∠R}\), which is \( =\frac{QR}{PQ}=\frac{5}{12}\).

Therefore, \( \text{ tan P} - \text{ cot R} = \frac{ 5}{12}-\frac{5}{12}=0\).