Question

Find the value of sin 2x+ cos 4x

Answer 1

Step-by-step explanation:

y=sin2(x)+cos4(x)————(1)y=sin2⁡(x)+cos4⁡(x)————(1)

=1−cos2(x)+cos4(x)=1−cos2⁡(x)+cos4⁡(x)

=1−cos2(x)(1−cos2(x))=1−cos2⁡(x)(1−cos2⁡(x))

=1−cos2(x)sin2(x)=1−cos2⁡(x)sin2⁡(x)

=1−4cos2(x)sin2(x)4=1−4cos2⁡(x)sin2⁡(x)4

=1−(2cos(x)sin(x))24=1−(2cos⁡(x)sin⁡(x))24

=1−sin2(2x)4————(2)=1−sin2⁡(2x)4————(2)

As −1≤sin(2x)≤1−1≤sin⁡(2x)≤1

⟹0≤sin2(2x)≤1⟹0≤sin2⁡(2x)≤1

⟹0≤sin2(2x)4≤1

Answer 2

Substituting cos²x = 1-sin²x in given expression, We get sin²x+(1-sin²x)² = sin²x + 1 -2sin²x +sin⁴x = 1 - sin²x + sin⁴x = 1 - sin²x(1-sin²x) = 1