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Find the values of theta in the interval...

Find the values of `theta` in the interval `(-pi/2,pi/2)` satisfying the equation `(1-tantheta)(1+tantheta)sec^2theta+2^tan^(2theta)=0`

Text Solution

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The correct Answer is:
`theta = pm pi //3`
Given, ` ( 1 - tan theta ) (1 + tan theta ) sec ^(2) theta + 2 ^(tan ^(2) theta ) = 0`
` rArr (1 - tan ^(2) theta ) * ( 1 + tan ^(2) theta ) + 2 ^(tan ^(2) theta ) = 0 `
` rArr 1- tan ^(4) theta + 2 ^(tan ^(2) theta = 0 `
Put ` " " tan ^(2) theta = x `
` therefore " " 1- x ^(2) + 2 ^(x) = 0 `
`rArr" " x ^(2) - 1 = 2 ^(x)`.
Note ` 2^(x) and x ^(2) - 1 ` are uncompatible functions, therefore we have to consider range to both functions.
Curve ` y = x ^(2)- 1 and y = 2 ^(x)`

It is clear from the graph that two curves intersect at one point at ` x = 3, y = 8`.
Therefore, ` tan ^(2) theta = 3`.
` rArr " " tan theta = pm sqrt3`.
` rArr" " theta = pm (pi)/(3)`
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