Page 44 - mathsvol1ch1to3ans

44

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◦
tan(75 ) + cot(75 ) = tan(45 + 30) + cot(75)
(tan(45) + tan(30)) cos(75)
= +
1 − tan(45) tan(30) sin(75)
√ √
!
1 6 − 2
1 + √ 4
3
= 1 + √ √ !
1 − √ 2 + 6
3 4
√ √ √
3 + 1 6 − 2
= √ + √ √
Not For Sale - Veeraragavan C S veeraa1729@gmail.com
3 − 1 6 + √ 2
√
= (2 + 3) + (2 − 3)
= 4
14. Prove that cos(A + B) cos C − cos(B + C) cos A = sin B sin(C − A).

Solution:
cos(A + B) cos(C) − cos(B + C) cos A = cos(C) cos(A) cos(B) − cos(C) sin(A) sin(B)

− cos(A)cos(B)cos(C) + cos(A) sin(B) sin(C)

= sin(B) (sin(C) cos(A) − sin(A) cos(C))
= sin(B) sin(C − A)

15. Prove that sin(n + 1)θ sin(n − 1)θ + cos(n + 1)θ cos(n − 1)θ = cos 2θ, n ∈ Z.

Solution:
sin(n + 1)θ sin(n − 1)θ + cos(n + 1)θ cos(n − 1)θ = cos ((n + 1) − (n − 1)) θ = cos 2θ

2π 4π
16. If x cos θ = y cos θ + = z cos θ + , [0.4cm] find the value of xy + yz + zx.
3 3
Solution:
1

Let x cos θ = y cos θ + 2π = z cos θ + 4π =
3 3 k

1 1 1 2π 4π
+ + = k cos θ + k cos θ + + k cos θ +
x y z 3 3
π
= k cos θ + 2k cos (θ + π) cos
3
= k cos θ + k cos (θ + π)
π π

= 2k cos θ + cos
2 2
= 0

1 1 1 yz + zx + xy
+ + =
x y z xyz

Hence xy+yz+zx = 0. ************************************************************************************************
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