Question

From the following figure, find the values of
(i) cos A 
(ii) cosec A
(iii) tan²A - sec²A 
(iv) sin C
(v) sec C 
(vi) cot² C - ` 1 / sin^2 "c"`

Answer 1

Consider the diagram as

Given angle  ADB = 90° and BDC = 90°
⇒ AB² = AD² + BD²         ...( AB is hypotenuse in ΔABD )

⇒ AB² = 3² + 4²

∴ AB² = 9 + 16 = 25 and AB = 5

⇒ BC² = BD² + DC²       ...( BC is hypotenuse in ΔBDC )

⇒ DC² = 12² - 4²

∴ DC² = 144 - 16 = 128 aand DC = 8`sqrt2` 

(i) cos A = `"base"/"hypotenuse" = "AD"/"AB" =3/5`

(ii) cosec A = `"hypotenuse"/"perpendicular" = "AB"/"BD" =5/4`

(iii) tan A = `"perpendicular"/"base" = "BD"/"AD" =4/3`

sec A = `"hypotenuse"/"base" = "AB"/"AD" =5/3`

tan² A - sec² A = `(4/3)^2 - (5/3)^2`

= `16/9 - 25/9`

= `( – 9)/(9)`

= – 1

(iv) sin C = `"perpendicular"/"hypotenuse" = "BD"/"BC" = 4/12 = 1/3`

(v) sec C = `"hypotenuse"/"base" = "BC"/"DC" = 12/(8sqrt2) = 3/(2sqrt2) = (3sqrt2)/4`

(vi) cot C = `"base"/"perpendicular" = "DC"/"BD" = (8sqrt2)/4 = 2sqrt2`

sin C = `"perpendicular"/"hypotenuse" = "BD"/"BC" = 4/12 = 1/3`

cot² C –  ` 1/sin^2C = (2sqrt2)^2 - 1/(1/3)^2`

= 8 - 9
=  – 1