Question

prove the above equal to 1/8​

Answer 1

Answer:

Step-by-step explanation:

sin(π/14) .sin(3π/14). Sin(5π/14)

//change some terms in cos .

sin(3π/14) = sin(π/2 - 4π/14) = cos(4π/14)

∴ sin(3π/14) = cos(4π/14)

Similarly, sin(5π/14) = sin(π/2 - 2π/14) = cos(2π/14)

=> sin(π/14).cos(4π/14).cos(2π/14)

//For ease of solving, Let π/14 = A

then, sin(π/14).cos(4π/14).cos(2π/14) = sinA.cos4A.cos2A

//multiply and divide by (2cosA)

= {2cosA.sinA.cos4A.cos2A}/(2cosA)

= Sin2A.cos4A.cos2A/(2cosA)

//We know, sin2A = 2sinA.cosA , so multiply numerator and denominator with 2

= {2sin2A.cos2A.cos4A}/(4cosA)

= sin4A.cos4A/(4cosA)

//We know, sin2A = 2sinA.cosA , so multiply numerator and denominator with 2

= (2sin4A.cos4A)/(8cosA)

= sin8A/8cosA

Put A = π/14

= sin8π/14/8cosπ/14

∵ sin8π/14 = sin(π/2 + π/14) = cosπ/14

=> sin8π/14/8cosπ/14 = cosπ/14/8cosπ/14 = 1/8

Hence proved.