Find the value of \sec(\pi/10). \sec( {3\pi}/10).\sec( {7\pi}/10).\sec( {9\pi}/10) =?

SSC CGL Model Paper 3

Question : 68 of 100

Marks: +1, -0

Find the value of

\sec(π/10). \sec( {3π}/10).\sec( {7π}/10).\sec( {9π}/10) =?

Solution:

\sec(π/10). \sec( {3π}/10).\sec( {7π}/10).\sec( {9π}/10) =?

⇒

1/x=\cosπ/10 .\cos {3π}/10 .\cos {7π}/10 .\cos {9π}/10

=\cosπ/10 .\cos {3π}/10(-\cos {3π}/10) (-\cos { π}/10)

=\cos^2π/10 .\cos ^2{3π}/10

=1/4(1+\cosπ/5)(1+\cos{3π}/5)

=1/4(1+\cosπ/5+\cos{3π}/5+\cos{ π}/5 .\cos{3π}/5)

=1/4(1+\cosπ/5+\cos{3π}/5+1/2(\cos{2 π}/5 .\cos{4π}/5))

=1/4(1+\cosπ/5+\cos{3π}/5-1/2(\cos{3 π}/5+\cos{ π}/5))

=1/4(1+1/2(\cos{ π}/5+\cos{ 3π}/5))

=1/4(1+1/2×(\cos36°+\cos108°))

=1/4(1+1/2×({√5+1}/4+\cos3(36°))

1/4(1+1/2×({√5+1}/4+4\cos^3 36°-3\cos36° ))

=1/4(1+1/2[{√5+1}/4+{4(√5+1)^3}/4^3

-{3×(√5+1)}/4])

=1/4(1+1/2[-1/2(√5+1)

+1/16(5√5+1+3.5+3√5)])

=1/4[1+1/4[-√5-1+1/8(8√5+16)]]

=1/4[1+1/4[2+√5-1√5]]

=5/4×1/4=5/16

⇒

x=16/5=3.2

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