Question

$\huge\fbox\colorbox{gold}{If..}$
$\small\fbox\colorbox{blue}{ tan A + cot A= 2}$
$\small\fbox\colorbox{red}{Prove:: tan¹⁰A + cot¹⁰A = 2}$​

Answer 1

Answer:

Explanation:

Given,

tanA + cotA = 2

To Prove :-

$tan^{10}A+cot^{10}A=2$

How To Do :-

Here they given a condition that 'tanA + cotA = 2' and asked us to prove that ' tan¹⁰A + cot¹⁰A = 2'. So by using that condition we need to change 'cotA' in terms of 'tanA' and we need to solve the quadratic equation which we got. After solving that quadratic equation we can get the value of 'tanA'. after obtaining the value of 'tanA' we can find the value of 'cotA' and we can find the value of 'tan¹⁰A + cot¹⁰A '

Formula Required :-

1) tanA = 1/cotA

2) cotA = 1/tanA

Solution :-

Taking the given condition :-

tanA + cotA = 2

tanA + 1/tanA = 2

[ ∴ cotA = 1/tanA ]

Taking L.C.M :-

[(tanA × tanA)+1]/tanA = 2

(tan²A + 1)/tanA = 2

tan²A + 1 = 2 × tanA

tan²A + 1 = 2tanA

tan²A - 2tanA + 1 = 0

Solving the quadratic equation :-

tan²A - tanA - tanA + 1 = 0

tanA(tanA - 1) - 1(tanA - 1) = 0

Taking '(tanA - 1)' as common :-

(tanA - 1)(tanA - 1) = 0

Equating both terms to zero :-

tanA - 1 = 0 , tanA - 1 = 0

∴ Both terms are same so we can equate any one of the term :-

tanA - 1 = 0

tanA = 1

∴ Value of 'tanA' = 1

→ cotA = 1/tanA

= 1/1

= 1

∴ Value of 'cotA' = 1

tan¹⁰A + cot¹⁰A :-

= (tanA)¹⁰ + (cotA)¹⁰

Substituting value of 'tanA' and 'cotA' :-

= (1)¹⁰ + (1)¹⁰

[ ∴ 1 power anything is '1' only ]

= 1 + 1

= 2

∴ tan¹⁰A + cot¹⁰A = 2.

Answer 2

Answer:

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