Correct Answer - Option 2 : 5/18
Given:
(cosD – secD)2 – (cosecD – sinD)2 = 0.6
Concepts used:
secD = 1/cosD
cosecD = 1/sinD
sin2D + cos2D = 1
sin2D = 2 × sinD × cosD
sin6D + cos6D = 1 – 3sin2Dcos2D
Calculation:
(cosD – secD)2 – (cosecD – sinD)2 = 0.6
⇒ {cosD – (1/cosD)}2 + {(1/sinD) – sinD}2 = 0.6
⇒ {(cos2D – 1)2/cos2D} + {(1 – sin2D)2/sin2D} = 0.6
⇒ {(1 – cos2D)2/cos2D} + {(1 – sin2D)2/sin2D} = 0.6
⇒ (sin2D)2/cos2D + (cos2D)2/sin2D = 0.6
⇒ sin4D/cos2D + cos4D/sin2D = 0.6
⇒ (sin6D + cos6D)/sin2Dcos2D = 0.6
⇒ (1 – 3sin2Dcos2D)/sin2Dcos2D = 0.6
⇒ 1 – 3sin2Dcos2D = 0.6sin2Dcos2D
⇒ 1 = 3.6sin2Dcos2D
⇒ sin2Dcos2D = 1/3.6
⇒ sin2Dcos2D = 10/36 = 5/18
∴ The value of sin2Dcos2D is 5/18