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Question

The number of distinct solutions of secx+tanx=3, where 0x3π is

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Solution

Here, secx+tanx=3
1+sinx=3cosx

3cosxsinx=1

dividing both sides by a2+b2 i.e 4=2 , we get
32cosx12sinx=12
cosπ6cosxsinπ6sinx=12
cos(x+π6)=12
cos(x+π6)=cosπ3
As 0x3π
π6x+π63π+π6
x+π6=π3,5π3,7π3
x=π6,3π2,13π6
But at x=3π2,tanx and secx is not defined.

Alternate Solution:
secx+tanx=3(1)
and we know that secxtanx=1secx+tanx=13(2)
From (1)+(2)
secx=23
From (1)(2)
tanx=13
tanx,secx both are positive only in 1st quadrant.
So, x=π6,13π6
Hence, the number of solutions is 2.

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