If-10sin-4-x-15cos-4-x-6-find-the-value-of-27cosec-6-x-8sec-6-x-

Question Number 195253 by Spillover last updated on 28/Jul/23

I f 10 \sin^{4} x + 15 \cos^{4} x = 6 .

f i n d t h e v a l u e o f

27 cosec^{6} x + 8 \sec^{6} x

Commented by Frix last updated on 28/Jul/23

250

Answered by BaliramKumar last updated on 28/Jul/23

{10 \sin}^{4} x + {15 \cos}^{4} x = 6

\frac{10}{6} \sin^{4} x + \frac{15}{6} \cos^{4} x = 1

\frac{5}{3} \sin^{4} x + \frac{5}{2} \cos^{4} x = 1

(\frac{5}{3} \sin^{2} x) \cdot \sin^{2} x + (\frac{5}{2} \cos^{2} x) \cdot \cos^{2} x = 1

(\frac{5}{3} \sin^{2} x) = 1 & (\frac{5}{2} \cos^{2} x) = 1

(\frac{5}{3}) = \frac{1}{\sin^{2} x} & (\frac{5}{2}) = \frac{1}{\cos^{2} x}

(\frac{5}{3}) = {cosec}^{2} x & (\frac{5}{2}) = \sec^{2} x

{27 cosec}^{6} x + {8 \sec}^{6} x

27 {(\frac{5}{3})}^{3} + 8 {(\frac{5}{2})}^{3} = 27 (\frac{125}{27}) + 8 (\frac{125}{8})

125 + 125 = 250

Commented by Spillover last updated on 03/Aug/23

t h a n k y o u

Answered by Spillover last updated on 02/Aug/23

10 \sin^{4} x + 15 \cos^{4} x = 6

d i v i d e b y {c o s}^{4} x b o t h s i d e s

10 \tan^{4} x + 15 = 6 \sec^{4} x

10 \tan \overset{4}{} x + 15 = 6 {(1 + \tan^{2} x)}^{2}

4 \tan^{4} x - 12 \tan^{2} x + 9 = 0

{(2 \tan^{2} x - 3)}^{2} = 0

\tan^{2} x = \frac{3}{2} \cot^{2} x = \frac{2}{3}

27 cosec^{6} x + 8 \sec^{6} x

27 {(1 + \cot^{2} x)}^{3} + 8 {(1 + \tan^{2} x)}^{3}

b u t \tan^{2} x = \frac{3}{2} \cot^{2} x = \frac{2}{3}

27 {(1 + \frac{2}{3})}^{3} + 8 {(1 + \frac{3}{2})}^{3}

27 {(\frac{5}{3})}^{3} + 8 {(\frac{5}{2})}^{3}

250