Question

2(sin6θ + cos6θ) − 3(sin4θ + cos4θ) is equal to
A. 0
B. 1
C. −1
D. None of these

Answer 1

Step-by-step explanation:

It's answer is -1 because this explanation is given in the pic see and get your answer

Answer 2

Given : 2(sin⁶θ + cos⁶θ) − 3(sin⁴θ + cos⁴θ)

solution :  

First, we take sin⁶ θ + cos⁶ θ = (sin²θ)³ + (cos²θ)³

By using this identity : (a + b)³ = a³ + b³ + 3a²b + 3ab²

From this identity we have : a³ + b³ = (a + b)³ – 3a²b – 3ab²

= (sin² θ + cos²θ)³ – 3 (sin²θ)² cos² θ – 3 sin²θ (cos² θ)²

= 1 -  3 sin⁴θ cos² θ – 3 sin²θ cos⁴θ

[ sin² θ + cos² θ = 1]

= 1 - 3 sin²θ cos²θ [sin² θ + cos² θ]

= 1 - 3 sin² θ cos² θ …………(1)

[sin²θ + cos²θ = 1 ]

 

Now, we take

sin⁴θ + cos⁴θ = (sin²θ)² + (cos²θ)²

By using this identity (a + b)² = a² + b² + 2ab

From this identity we have : a² + b² = (a + b)² – 2ab

= (sin²θ + cos²θ)² – 2 sin²θ cos²θ

= 1 - 2 sin²θ cos²θ …………….(2)

[sin² θ + cos² θ = 1]

Now, using eq (1) and (2), we have:  

2(sin⁶θ + cos⁶θ) − 3(sin⁴θ + cos⁴θ)

= 2(1 - 3 sin² θ cos² θ) – 3(1 - 2 sin²θ cos²θ)

= 2 - 6 sin² θ cos² θ - 3 + 6 sin² θ cos² θ

= 2 - 3  

= - 1

2(sin⁶θ + cos⁶θ) − 3(sin⁴θ + cos⁴θ) = - 1

Hence, 2(sin⁶θ + cos⁶θ) − 3(sin⁴θ + cos⁴θ)  is equal to - 1.

Among the given options option (C) - 1 is correct.

HOPE THIS ANSWER WILL HELP YOU……

 

Some more questions :  

If x = a sec θ and y = b tan θ, thenb²x²-a²y² =

(a)ab

(b)a² − b²

(c)a² + b²

(d)a² b²

https://brainly.in/question/11828258

 

If x = a cos θ and y = b sin θ, then b²x²=a²y² =

(a)a² b²

(b)ab

(c)a⁴ b⁴

(d)a² + b²

https://brainly.in/question/11828193