Find the value of sin54÷sin18-cos54÷cos18
Answers
please explain this in details
Answer:
Step-by-step explanation:
Let x = 18 degrees
sin (54) = sin (3x)
cos (54)= cos (3x)
[By using the formula : sin 3x = 3 sin x - 4 sin³ x ]
sin54 ÷ sin18 = sin 3x ÷ sin x
= (3 sin x - 4 sin³ x) / sin x
= 3 - 4 sin² x __________ Equation 1
Similarly for other part of the equation :
[By using the formula : cos 3x = 4 cos³ x - 3 cos x ]
cos54 ÷ cos18 = cos 3x ÷ cos x
= (4 cos³ x - 3 cos x) / cos x
= 4 cos² x - 3 __________ Equation 2
Subtracting equation 2 from 1 :
sin54÷sin18-cos54÷cos18
= ( 3 - 4 sin² x ) - (4 cos² x - 3 )
= 3 - 4 sin² x - 4 cos² x + 3
= 6 - 4 (sin² x + cos² x)
= 6 - 4 = 2
{ Note : For whatever value of the angle , the answer to this type of question is 2 , since we have not substituted any trigonometric value of 18 degree in any equation }