Question

if 10 sin⁴ alpha+ 15 cos⁴alpha = 6 , find the value of 27 cosec^6 alpha + 8 sec ^6 alpha.​

Answer 1

We have,

10 sin⁴ alpha + 15 cos⁴ alpha = 6.(1)²

10 sin⁴ alpha + 15 cos⁴ alpha = 6 (sin²alpha + cos² alpha)²

[1 = sin² theta + cos² theta ]

Dividing both sides by cos⁴ alpha.

10 sin⁴alpha/cos⁴ alpha + 15 cos⁴alpha /cos⁴ alpha = 6 ( sin²aplha/cos²alpha + cos²alpha/ cos²alpha)⁴

10 tan⁴alpha +15 = 6 (tan²alpha + 1)²

10 tan⁴alpha + 15 = 6(tan⁴alpha +1+2tan²alpha)

[(a+b)²= a²+b² +2ab]

10tan⁴alpha + 15 = 6 tan⁴alpha +6 +12tan²alpha.

4tan⁴alpha -12 tan²alpha+9 = 0

(2tan²alpha -3)²= 0

2tan²alpha -3 = 0

tan²alpha = 3/2. So,

cot²alpha = 2/3

Now , 27( cosec²alpha)³ +8( sec²alpha)³

27 ( 1+cot²alpha)³ + 8 (1+tan²alpha)³

27+(1+2/3)³+8(1+3/2)³

27×125/27+8×125/8

125 +125

250

Answer 2

SOLUTION

Refer to the attachment

hope it helps ☺️