Question

if sin⁴x/2 + cos⁴x/3 =1/5 find sin⁸x/8 + cos⁸x/27 = 125​

Answer 1

Answer:

sin

4

x

+

3

cos

4

x

=

5

1

2

(sin

2

x)

2

+

3

(1−sin

2

x)

2

=

5

1

Let sin

2

x be t,t≥0

t∈[0,1]

2

t

2

+

3

(1−t)

2

=

5

1

2

t

2

+

3

1−2t+t

2

=

5

1

6

3t

2

+2−4t+2t

2

=

5

1

25t

2

−20t+10=6

25t

2

−20t+4=0

25t

2

−10t−10t+4=0

(5t−2)(5t−2)=0

t=

5

2

∴sin

2

x=

5

2

cos

2

x=1−sin

2

x=1−

5

2

=

5

3

tan

2

x=

cos

2

x

sin

2

x

=

3

2