Math, asked by samanviakshaj, 5 months ago

3(sin x + cos x)4 + 6 (sin x- cos x)2 + 4(sinⓇx+cosy
2) 11
3) 12
4) 13
1) 10​

Answers

Answered by MisterIncredible
23

Question : -

3 (sin x - cos x)⁴ + 6 (sin x + cos x)² + 4 (sin⁶ x + cos⁶ x) = ?

ANSWER

SOLUTION : -

3 (sin x - cos x)⁴ + 6 (sin x + cos x)² + 4 (sin⁶ x + cos⁶ x)

Here, let's solve this first part by part !

3 (sin x - cos x)⁴

3 ([sin x - cos x]²)²

Using the identity;

  • (a - b)² = +-2ab

3 ([sin² x + cos² x - 2 sin x cos x])²

  • sin² x + cos² x = 1

3 (1 - 2 sin x cos x)²

3 [(1)² + (2 sin x cos x)² - 2(1)(2 sin x cos x) ]

3 [1 + 4 sin² x cos² x - 4 sin x cos x]

3 + 12 sin² x cos² x - 12 sin x cos x »»(1)

Now,

6 (sin x + cos x)²

  • (a + b)² = ++2ab

6 (sin² x + cos² x + 2 sin x cos x)

6 (1 + 2 sin x cos x)

6 + 12 sin x cos x »» (2)

Similarly,

4 (sin⁶ x + cos⁶ x)

4 ([sin² x]³ + [cos² x]³)

  • + = (a + b) ( - ab + )

Here,

  • a = sin² x
  • b = cos² x

4 (sin² x + cos² x) ([sin² x]² - [sin² x][cos² x] + [cos² x]²

4 (1) (sin⁴ x - sin² x cos² x + cos⁴ x)

4 (sin⁴ x - sin² x cos² x + cos⁴ x)

4sin⁴ x - 4 sin² x cos² x + 4cos⁴ x »»(3)

Adding (1),(2) & (3)

3 + 12 sin² x cos² x - 12 sin x cos x + 6 + 12 sin x cos x + 4sin⁴ x - 4 sin² x cos² x + 4cos⁴ x

3 + 12 sin² x cos² x + 6 + 4 sin⁴ x - 4 sin² x cos² x + 4 cos⁴ x

9 + 8 sin² x cos² x + 4 sin⁴ x + 4 cos⁴ x

9 + 4([sin⁴ x + cos⁴ x + 2 sin² x cos² x ])

9 + 4([sin² x]² + [cos² x]² + 2 [sin² x] [cos² x])

  • a² + b² + 2ab = (a+b)²

9 + 4([sin² x + cos² x]²)

9 + 4([1]²)

9 + 4(1)

9 + 4

= 13

Answered by 2008shrishti
1

Question : -

3 (sin x - cos x)⁴ + 6 (sin x + cos x)² + 4 (sin⁶ x + cos⁶ x) = ?

ANSWER

SOLUTION : -

3 (sin x - cos x)⁴ + 6 (sin x + cos x)² + 4 (sin⁶ x + cos⁶ x)

Here, let's solve this first part by part !

3 (sin x - cos x)⁴

3 ([sin x - cos x]²)²

Using the identity;

(a - b)² = a²+b²-2ab

3 ([sin² x + cos² x - 2 sin x cos x])²

sin² x + cos² x = 1

3 (1 - 2 sin x cos x)²

3 [(1)² + (2 sin x cos x)² - 2(1)(2 sin x cos x) ]

3 [1 + 4 sin² x cos² x - 4 sin x cos x]

3 + 12 sin² x cos² x - 12 sin x cos x »»(1)

Now,

6 (sin x + cos x)²

(a + b)² = a²+b²+2ab

6 (sin² x + cos² x + 2 sin x cos x)

6 (1 + 2 sin x cos x)

6 + 12 sin x cos x »» (2)

Similarly,

4 (sin⁶ x + cos⁶ x)

4 ([sin² x]³ + [cos² x]³)

a³ + b³ = (a + b) (a² - ab + b²)

Here,

a = sin² x

b = cos² x

4 (sin² x + cos² x) ([sin² x]² - [sin² x][cos² x] + [cos² x]²

4 (1) (sin⁴ x - sin² x cos² x + cos⁴ x)

4 (sin⁴ x - sin² x cos² x + cos⁴ x)

4sin⁴ x - 4 sin² x cos² x + 4cos⁴ x »»(3)

Adding (1),(2) & (3)

3 + 12 sin² x cos² x - 12 sin x cos x + 6 + 12 sin x cos x + 4sin⁴ x - 4 sin² x cos² x + 4cos⁴ x

3 + 12 sin² x cos² x + 6 + 4 sin⁴ x - 4 sin² x cos² x + 4 cos⁴ x

9 + 8 sin² x cos² x + 4 sin⁴ x + 4 cos⁴ x

9 + 4([sin⁴ x + cos⁴ x + 2 sin² x cos² x ])

9 + 4([sin² x]² + [cos² x]² + 2 [sin² x] [cos² x])

a² + b² + 2ab = (a+b)²

9 + 4([sin² x + cos² x]²)

9 + 4([1]²)

9 + 4(1)

9 + 4

= 13

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