Question

Express the following as the product of sines and cosines : sin 2x + cos 4x​

Answer 1

Trigonometry

Given: sin2x + cos4x

To find: express the sum as the product of sines and cosines

Solution:

• We know that, sin2A = 2 sinA cosA and cos2A = cos²A - sin²A

• Now, sin2x + cos4x
• = 2 sinx cosx + cos2(2x)
• = 2 sinx cosx + cos²2x - sin²2x
• = 2 sinx cosx + (cos²x - sin²x)² - (2 sinx cosx)²
• = 2 sinx cosx + (cos²x + sin²x)² - 4 cos²x sin²x - 4 sin²x cos²x
• = 2 sinx cosx + 1 - 8 sin²x cos²x
• = 1 + 2 sinx cosx - 8 (sinx cosx)²
• This is the required expression.

Answer:

sin2x + cos4x = 1 + 2 sinx cosx - 8 (sinx cosx)²