Question

Question 12 Prove that sin2 6x – sin2 4x = sin 2x sin 10x

Class X1 - Maths -Trigonometric Functions Page 73

Answer 1

LHS = sin²(6x) - sin²(4x)
Use the formula,
(a² - b²) = (a - b)(a + b)
= (sin6x -sin4x)(sin6x +sin4x)
Now,
Use the formula,
sinC + sinD = 2sin(C+D)/2.cos(C-D)/2
sinC - sinD = 2cos(C+D)/2.sin(C-D)/2

={2cos(6x+4x)/2.sin(6x-4x)/2}{2sin(6x+4x)/2.cos(6x-4x)/2}
={2cos5x.sinx}{2sin5x.cosx}
={2sin5x.cos5x}{2sinx.cosx}

Use the formula,
sin2A = 2sinA.cosA

= sin2(5x).sin2(x)
= sin10x.sin2x =RHS

Answer 2

sin²6x-sin²4x

=(sin6x+sin4x)( sin6x-sin4x)

=2sin(4x+6x/2)cos(6x-4x/2)*2cos(4x+6x/2)sin(6x-4x/2)

=2sin5xcosx*2cos5x sinx

=2cos5x*sin5x(2cosx*sinx)

=sin2(5)x*sin2(1)x

=sin2x*sin10x

Formula :-
a+b ( a-b) = a^2-b^2

sin(c+d) = 2 sin( c + d /2) cos (c-d/2)

sin(c-d) = 2 cos( c + d /2) sin (d-c/2)

sin2∅ = 2sin∅ cos∅