Trigonometry
no. of solutions of the equation tan^2x-sec^10x+1=0 in (0,10) ?
no. of solutions of the equation tan^2x-sec^10x+1=0 in (0,10) ?
1 Answer
Y RAJYALAKSHMI
tan2x – sec10x + 1 = 0
=> sec2x – sec10x = 0 (as 1+ tan2x = sec2x)
=>sec2x(1 – sec8x) = 0
=>sec2x(1 – secx)(1 + secx)(1 + sec2x)(1 + sec4x) = 0
=> sec2x = 0, (1 – secx)= 0 , (1 + secx) = 0, (1 + sec2x)= 0, (1 + sec4x) = 0
The real roots are only for sec2x = 0, (1 – secx)= 0 , (1 + secx) = 0
sec2x = 0 => secx = 0
(1 – secx)= 0 => secx = 1
(1 + secx) = 0 => secx = – 1
only one solution secx = 1 is in the interval (0, 10)
Ans = 1
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