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
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
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