If A = 30°; show that : cos 2A = cos4 A - sin4 A

cos 2A = cos4 A - sin4 A L.H.S. = cos 2A = cos (2 x 30°) = cos 60° = ∴ L.H.S. = R.H.S. Hence, cos 2A = cos4 A - sin4 A.

Mathematics

If A = 30°; show that :
cos 2A = cos⁴ A - sin⁴ A

Trigonometric Identities

3 Likes

Answer

cos 2A = cos⁴ A - sin⁴ A

L.H.S. = cos 2A

= cos (2 x 30°)

= cos 60°

= $\dfrac{1}{2}$

$\text{R.H.S.} = \text{cos}^4 A - \text{sin}^4 A\\[1em] = \text{cos}^4 30° - \text{sin}^4 30°\\[1em] = \Big(\dfrac{\sqrt3}{2}\Big)^4 - \Big(\dfrac{1}{2}\Big)^4\\[1em] = \dfrac{9}{16} - \dfrac{1}{16}\\[1em] = \dfrac{9 - 1}{16}\\[1em] = \dfrac{8}{16}\\[1em] = \dfrac{1}{2}$

∴ L.H.S. = R.H.S.

Hence, cos 2A = cos⁴ A - sin⁴ A.

Answered By

1 Like

Mathematics

If A = 30°; show that :
cos 2A = cos⁴ A - sin⁴ A

Trigonometric Identities

Answer

Related Questions

If A = 30°; show that :
sin 3 A = 4 sin A sin (60° - A) sin (60° + A)

If A = 30°; show that :
(sin A - cos A)² = 1 - sin 2 A

If A = 30°; show that :
$\dfrac{\text{1 - cos 2 A}}{\text{sin 2 A}}$ = tan A

If A = 30°; show that :
$\dfrac{\text{1 + sin 2 A + cos 2 A}}{\text{sin A + cos A}}$ = 2 cos A

Mathematics

If A = 30°; show that :cos 2A = cos4 A - sin4 A

Trigonometric Identities

Answer

Related Questions

If A = 30°; show that :sin 3 A = 4 sin A sin (60° - A) sin (60° + A)

If A = 30°; show that :(sin A - cos A)2 = 1 - sin 2 A

If A = 30°; show that :1 - cos 2 Asin 2 A\dfrac{\text{1 - cos 2 A}}{\text{sin 2 A}}sin 2 A1 - cos 2 A​ = tan A

If A = 30°; show that :1 + sin 2 A + cos 2 Asin A + cos A\dfrac{\text{1 + sin 2 A + cos 2 A}}{\text{sin A + cos A}}sin A + cos A1 + sin 2 A + cos 2 A​ = 2 cos A

If A = 30°; show that :
cos 2A = cos⁴ A - sin⁴ A

If A = 30°; show that :
sin 3 A = 4 sin A sin (60° - A) sin (60° + A)

If A = 30°; show that :
(sin A - cos A)² = 1 - sin 2 A

If A = 30°; show that :
$\dfrac{\text{1 - cos 2 A}}{\text{sin 2 A}}$ = tan A

If A = 30°; show that :
$\dfrac{\text{1 + sin 2 A + cos 2 A}}{\text{sin A + cos A}}$ = 2 cos A