Question

if sin theta+cosec theta=2 then find the value of sin^(10)theta+cosec^(10)theta

Answer 1

ANSWER:-

sinθ + cosecθ = 2

Simplifying we get

sinθ + 1/sinθ = 2

sin²θ + 1/sinθ = 2

sin²θ + 1 = 2sinθ

sin²θ - 2sinθ + 1 = 0

→ sin²θ - 2(sinθ)(1) + 1²

It is in the form of a² - 2ab + b² = (a - b)²

→ (sinθ - 1)² = 0

→ sinθ - 1 = 0

→ sinθ = 1

Using

sinθ = 1/cosecθ

→ 1/cosecθ = 1

→ cosecθ = 1

Hence sinθ = 1 & cosecθ = 1

Substituting the values

sin¹⁰θ + cosec¹⁰θ

→ 1¹⁰ + 1¹⁰

→ 1 + 1

Hence, sin¹⁰θ + cos¹⁰θ = 2

Answer 2

ANSWER :-

↦ sinθ + cosecθ = 2

Simplifying we get

↦ + 1/sinθ = 2

↦ sinθsin²θ + 1/sinθ = 2

↦ sin²θ + 1 = 2sinθ

↦ sin²θ - 2sinθ + 1 = 0

↦ sin²θ - 2(sinθ)(1) + 1²

By using ( a - b )² = a² - 2ab + b²

→ (sinθ - 1)² = 0

→ sinθ - 1 = 0

→ sinθ = 1

Using

sinθ = 1/cosecθ

↦ 1/cosecθ = 1

↦ cosecθ = 1

Hence sinθ = 1 & cosecθ = 1

Substituting the values

sin¹⁰θ + cosec¹⁰θ

↦ 1¹⁰ + 1¹⁰

↦ 1 + 1

Hence, sin¹⁰θ + cos¹⁰θ = 2