cos 15° - sin 15º = 1/root2
Answers
Answer:
√2 / 2
Step-by-step explanation:
To prove --->
Cos15° - Sin15° = 1 /√2
Proof---> We know that,
Cos ( A - B ) = CosA CosB + SinA SinB
Putting A = 45° and B = 30° , we get,
Cos( 45° - 30° ) = Cos45° Cos30° + Sin45° Sin30°
=> Cos15° = Cos45° Cos30° + Sin45° Sin30°
Putting values , we get,
= ( 1 / √2 ) ( √3 / 2 ) + ( 1 / √2 ) ( 1 / 2 )
= ( √3 / 2√2 ) + ( 1 / 2√2 )
Taking 2√2 as LCM , we get,
=> Cos15° = ( √3 + 1 ) / 2√2
Now , we know that,
Sin( A - B ) = SinA CosB - CosA SinB
Putting A = 45° and B = 30° , we get,
Sin( 45° - 30° ) = Sin45° Cos30° - Cos45° Sin30°
Putting values , we get,
=> Sin15° = ( 1 / √2 ) ( √3 / 2 ) - ( 1 / √2 ) ( 1 / 2 )
=> Sin15° = ( √3 / 2√2 ) - ( 1 / 2√2 )
Taking 2√2 as LCM , we get,
=> Sin15° = ( √3 - 1 ) / 2√2
Now , returning to original problem,
Cos15° - Sin15°
= { (√3 + 1 ) / 2√2 } - { (√3 - 1 ) / 2√2 }
Taking 2√2 as LCM, we get,
= { ( √3 + 1 ) - ( √3 - 1 ) } / 2√2
= ( √3 + 1 - √3 + 1 ) / 2√2
√3 and - √3 cancel out each other, and we get,
= 2 / 2√2
= 1 / √2
= 1 × √2 / √2 × √2
= √2 / 2
#Answerwithquality
#BAL
√2/2
#answerwithquality #bal