Question

If tan theta +cot theta=2 find tan 7​ theta +cot 7 theta

Answer 1

it is given that, tanθ + cotθ = 2

⇒tanθ + 1/tanθ = 2

⇒ tan²θ + 1 = 2tanθ

⇒tan²θ - 2tanθ + 1 = 0

⇒(tanθ - 1)² = 0

⇒tanθ = 1

so, cotθ = 1/tanθ = 1

then, $tan^7\theta+cot^7\theta$

= $(tan\theta)^7+(cot\theta)^7$

= $(1)^7+(1)^7$

= 1 + 1

= 2

hence, $tan^7\theta+cot^7\theta=2$

short trick :

you can solve this type of question using hit and trial method

tanθ + cotθ = 2, you can assume θ = 45°

as we know, tan45° = 1 and cot45° = 1

so, tan45° + cot45° = 2 (satisfied)

now apply it in tan^7θ + cot^7θ

= (tanθ)^7 + (cotθ)^7

= 1^7 + 1^7 = 1 + 1 = 2 [Ans]

Answer 2

Step-by-step explanation:

my answer is correct 100 %