Question

if tan a + cot a = 3, find value of ( tan^2a + cot^2 a) *( tan^3a + cot^3a)​

Answer 1

Answer :-

126

Solution :-

1. Finding tan²a + cot²a

tan a + cot a = 3

Squaring on both sides

⇒ (tan a + cot a)² = 3²

⇒ tan² a + cot² a + 2 * tan a * cot a = 9

[ Because (x + y)² = x² + y² + 2xy ]

⇒ tan² a + cot² a + 2 * tan a * 1/tan a = 9

[ Because cot a = 1/tan a ]

⇒ tan² a + cot² a + 2 = 9

⇒ tan² a + cot² a = 9 - 2 = 7

2. Finding tan³a + cot³a

tan a + cot a = 3

Cubing on both sides

⇒ (tan a + cot a)³ = 3³

⇒ tan³ a + cot³ a + 3 * tan a * cot a * (tan a + cot a) = 27

[ Because (x + y)³ = x³ + y³ + 3xy(x + y) ]

⇒ tan³ a + cot³ a + 3 * tan a * 1/tan a * 3 = 27

[ Because tan a + cot a = 3 ]

⇒ tan³ a + cot³ a + 9 = 27

⇒ tan³ a + cot³ a = 27 - 9 = 18

3. Finding (tan² a + cot² a) * (tan³ a + cot³ a)

(tan² a + cot² a) * (tan³ a + cot³ a) = 7 * 18 = 126

Therefore the value of (tan² a + cot² a) * (tan³ a + cot³ a) is 126.