Question

Prove that:-2cos^2theta-cos^4theta+sin^4theta=1​

Answer 1

Required Answer:-

Given to prove:

• 2cos²θ - cos⁴θ + sin⁴θ = 1

Proof:

Taking LHS,

2cos²θ - cos⁴θ + sin⁴θ

= 2cos²θ + sin⁴θ - cos⁴θ

= 2cos²θ + (sin²θ)² - (cos²θ)²

= 2cos²θ + (sin²θ + cos²θ)(sin²θ - cos²θ)

We know that,

➡ sin²θ + cos²θ = 1

Therefore,

2cos²θ + (sin²θ + cos²θ)(sin²θ - cos²θ)

= 2cos²θ + sin²θ - cos²θ

= sin²θ + cos²θ

= 1 [As sin²θ + cos²θ = 1]

= RHS (Hence Proved)

Formula Used:

• sin²θ + cos²θ = 1

More Formulae:

• cosec²θ - cot²θ = 1
• sec²θ - tan²θ = 1
• sin(90° - θ) = cosθ
• cosec(90° - θ) = secθ
• tan(90° - θ) = cotθ
• sinθ/cosθ = tanθ