Question

sin^4 theta - cos^4 theta = 1 - 2 cos^2 theta​

Answer 1

EXPLANATION.

⇒ sin⁴θ - cos⁴θ = 1 - 2cos²θ.

As we know that,

We can write equation as,

⇒ (sin²θ)² - (cos²θ)².

As we know that,

Formula of :

⇒ (a² - b²) = (a + b)(a - b).

⇒ [sin²θ + cos²θ][sin²θ - cos²θ].

⇒ 1[sin²θ - cos²θ].

As we know that,

Formula of :

⇒ sin²θ + cos²θ = 1.

⇒ sin²θ = 1 - cos²θ.

⇒ [1 - cos²θ - cos²θ].

⇒ [1 - 2cos²θ].

Hence proved.

                                                                                                                       

MORE INFORMATION.

Inverse trigonometric ratios of multiple angles.

(1) = 2sin⁻¹x = sin⁻¹(2x√1 - x²), if -1 ≤ x ≤ 1.

(2) = 2cos⁻¹x = cos⁻¹(2x² - 1), if -1 ≤ x ≤ 1.

(3) = 2tan⁻¹x = tan⁻¹(2x/1 - x²) = sin⁻¹(2x/1 + x²) = cos⁻¹(1 - x²/1 + x²) = 2tan⁻¹x (|x| < 1) = π - 2tan⁻¹x,(|x| > 1).

(4) = 3sin⁻¹x = sin⁻¹(3x - 4x³).

(5) = 3cos⁻¹x = cos⁻¹(4x³ - 3x).

(6) = 3tan⁻¹x = tan⁻¹(3x - x³/1 - 3x²).

Answer 2

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⇒ sin⁴θ - cos⁴θ = 1 - 2cos²θ.

As we know that,

We can write equation as,

⇒ (sin²θ)² - (cos²θ)².

As we know that,

Formula of :

⇒ (a² - b²) = (a + b)(a - b).

⇒ [sin²θ + cos²θ][sin²θ - cos²θ].

⇒ 1[sin²θ - cos²θ].

As we know that,

Formula of :

⇒ sin²θ + cos²θ = 1.

⇒ sin²θ = 1 - cos²θ.

⇒ [1 - cos²θ - cos²θ].

⇒ [1 - 2cos²θ].

Hence proved.

                                                                                                                       

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