Simplify y = (sin 4x).(cos4x)

The simplified value of y = (sin 4x).(cos4x) is 1/2 sin(8x). This is solved using trigonometric formulas and a detailed explanation for the same is added below:

Simplify: y = sin 4x × cos4x

Solution:

y = sin 4x.cos4x

y = 2/2{sin 4x.cos4x}

y = 1/2{2sin 4x.cos4x}

Using Double Angle Formulas (sin2A = 2sinAcosA)

y = 1/2 sin(8x)

Sample Problems

Problem 1: Simplify: 12m² - 9m + 5m - 4m² - 7m + 10.

Solution:

12m² - 9m + 5m - 4m² - 7m + 10

= (12 - 4)m² + (5 - 9 - 7)m + 10

= 8m² + (-4 - 7)m + 10

= 8m² + (-11)m + 10

= 8m² - 11m + 10

Problem 2: Find the degree of the monomial 7.

Solution: 

Degree of any constant term is zero (0). So, the degree of the monomial 7 is 0.

Problem 3: Simplify y = (sin 8x).(cos8x)

Solution: 

y = sin 8x.cos8x

y = 2/2{sin 8x.cos8x}

y = 1/2{2sin 8x.cos8x}

Using Double Angle Formulas (sin2A = 2sinAcosA)

y = 1/2 sin(16x)

Problem 4: Simplify cos(3x). cos(4x)

Solution:

We know that, cos A cos B = 1/2{cos(A+B) + cos(A-B)}

= cos(3x). cos(4x)

Here, A = 3x and B = 4x

= 1/2{cos(3x+4x) + cos(3x-4x)}

= 1/2{cos(7x) + cos(-x)}

Using cos(-x) = cos(x)

= 1/2{cos(7x) + cos(x)}