Prove that (sin⁴ θ - cos⁴ θ + 1) cosec² θ = 2

Solution:

We know that

LHS = (sin⁴ θ - cos⁴ θ + 1) cosec² θ

Using the algebraic identity

a² - b² = (a + b) (a - b)

We can write sin⁴ θ - cos⁴ θ = (sin² θ + cos² θ) (sin² θ - cos² θ)

= [(sin² θ + cos² θ) (sin² θ - cos² θ) + 1] cosec² θ

We know that

sin² θ + cos² θ = 1

By substituting it we get

= [sin² θ - cos² θ + 1] cosec² θ

As sin² θ + cos² θ = 1

We can write it as

1 - cos² θ = sin² θ

So we get

= 2 sin² θ cosec² θ

Here

= 2 sin² θ . 1/sin² θ

= 2

= RHS

Therefore, it is proved.

✦ Try This: Prove that: 2sec² θ - sec⁴ θ - 2cosec² θ + cosec⁴ θ = cot⁴ θ - tan⁴ θ.

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 8

NCERT Exemplar Class 10 Maths Exercise 8.3 Sample Problem 2

Prove that (sin⁴ θ - cos⁴ θ + 1) cosec² θ = 2

Summary:

Trigonometry is a branch of mathematics that deals with the relation between the angles and sides of a right-angled triangle. It is proved that (sin⁴ θ - cos⁴ θ + 1) cosec² θ = 2

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