Question

Which of the following expressions is equivalent to sin 6x :
sin 2x cos 4x - cos 4x sin 2x sin 2x cos 4x + cos 2x sin 4x sin 2x + cos (4x) sin 2x + sin 4x None of the above​

Answer 1

y=sin2(x)+cos4(x)————(1)

=1−cos2(x)+cos4(x)

=1−cos2(x)(1−cos2(x))

=1−cos2(x)sin2(x)

=1−4cos2(x)sin2(x)4

=1−(2cos(x)sin(x))24

=1−sin2(2x)4————(2)

As −1≤sin(2x)≤1

⟹0≤sin2(2x)≤1

⟹0≤sin2(2x)4≤14

⟹0≥−sin2(2x)4≥−14

⟹1−14≤1−sin2(2x)4≤1

⟹34≤y≤1

So, range of sin2(x)+cos4(x) is [ 34 , 1 ]

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Answer 2

Answer:

Given, p=sin

2

x+cos

4

x

=sin

2

x+cos

2

x(1−sin

2

x)

=(sin

2

x+cos

2

x)−sin

2

xcos

2

x

=1−sin

2

xcos

2

x≤1....(i)

Again, p=1−cos

2

x+cos

4

x

=(cos

2

x−

2

1

)

2

+

4

3

⇒p≥

4

3

....(ii)

∴ From Eqs. (i) and (ii), we get

4

3

≤p≤1.

Step-by-step explanation:

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