# In Fig. 8.13, find tan P – cot R

In the given triangle PQR, the given triangle is right-angled at Q and the given measures are:

In the given triangle PQR, the given triangle is right-angled at Q and the given measures are:

PR = 13cm,

PQ = 12cm

Since the given triangle is a right-angled triangle, to find the side QR, apply the Pythagorean theorem. According to the Pythagorean theorem, in a right-angled triangle, the squares of the hypotenuse side are equal to the sum of the squares of the other two sides.

PR2 = QR2 + PQ2

**Substitute the values of PR and PQ**

132 = QR2+122

169 = QR2+144

**Therefore, QR2 = 169−144**

QR2 = 25

QR = √25 = 5

**Therefore, the side QR = 5 cm**

**To find tan P – cot R:**

According to the trigonometric ratio, the tangent function is equal to the ratio of the length of the opposite side to the adjacent sides, the value of tan (P) becomes **tan (P) = Opposite side /Adjacent side = QR/PQ = 5/12**

Since the Cot function is the reciprocal of the tan function, the ratio of cot function becomes,

Cot (R) = Adjacent side/Opposite side = QR/PQ = 5/12

Therefore, tan (P) – cot (R) = 5/12 – 5/12 = 0

Therefore, tan(P) – cot(R) = 0