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Question

If 10sin4x+15cos4x=6, then the numerical value of 24sec6x27 cosec6 x is

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Solution

10sin4x+15cos4x=610sin4x+15cos4x=6×(1)210sin4x+15cos4x=6(sin2x+cos2x)2
Dividing both sides by cos4x we get
10tan4x+15=6(tan2x+1)24tan4x12tan2x+9=0(2tan2x3)2=0tan2x=32cot2x=23
Now,
24sec6x27 cosec6 x=24(sec2x)327( cosec2 x)3=24(1+tan2x)327(1+cot2x)3=24(52)327(53)3=53(31)=250

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