Find the value of cos (a), if cos (a) ^4 - sin (a) ^4 = 1/8

cos a = 3/4

Step-by-step explanation:

Step 1: Given details are cos (a) ^4 - sin (a) ^4 = 1/8

Now, cos (a) ^4 can also be written as (cos²a) ² and sin (a) ^4 can be written as (sin²a) ²

⇒ (cos²a) ² - (sin²a) ² = 1/8

Step 2: Apply the formula for a² - b² = (a - b) (a + b). Here a = cos²a and b = sin²a

⇒ (cos²a) ² - (sin²a) ² = (cos²a - sin²a) (cos²a + sin²a) = 1/8

⇒ (cos²a - sin²a) = 1/8 since cos²a + sin²a = 1

⇒ cos²a - (1 - cos²a) = 1/8 since sin²a = 1 - cos²a

⇒ 2 cos²a - 1 = 1/8

⇒ 2 cos²a = 1 + 1/8 = 9/8

⇒ cos²a = 9/16

⇒ cos a = 3/4

cosa = ±3/4

Step-by-step explanation:

cos (a) ^4 - sin (a) ^4 = 1/8

(cos (a) ² - sin (a) ²) * (cos (a) ² + sin (a) ²) = 1/8; {remember: cos (a) ² + sin (a) ² = 1}

(cos (a) ² - sin (a) ²) = 1/8

cos (a) ² - (1 - (cos (a) ² = 1/8

2cos (a) ² = 9/8

2cos (a) ² = 9/16

cosa = ±3/4