Question

Please give correct solution

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

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Answer 1

Given:

Tan a + cot a = 3

To find:

(Tan²a + cot ² a) (Tan ³a + cot ³a) =?

Formula used :

(a + b) ² = a² + b² + 2ab

(a + b)³ = a³+ b³+ 3ab(a + b)

Solution:

As it is given that tan a + cot a = 3, then

➨ (tan a+cot a)² = tan²a +cot ²a +2×tan a×cot a

As cot a = 1/tan a, so tan a × cot a = 1.

➨ 3² = tan²a+ cot² a +2

➨ 9 = tan²a + cot²a +2

➨ 9-2 = tan²a + cot ²

➨ 7 = tan²a + cot² a ........................ Eq 1

And,

➨ (tan a+cot a) ³ = tan³a +cot ³a +3×tan a×cot a(tan a +cot a)

➨ (3)³ = tan³a + cot ³ a +3 × 1 ×(3)

➨ 27 = tan³a + cot ³ a +9

➨ 27-9 = tan³a + cot³a

➨ 18 = tan³a + cot ³ a ..................... Eq 2

Hence from equation 1 & 2 we can conclude that, (tan²a + cot²a) (tan³a + cot³a) = 7 × 18 =126.

Therefore:-

The required answer is 126.