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The number of solutions of the equation 8tan2θ+9=6secθ in the interval (π2,π2).
A.Two
B.Four
C.Zero
D.None of these

Answer
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Hint: Here, we have to find the number of solutions. First, we have to solve the given equation to find the number of solutions. Trigonometric equation is an equation involving one or more trigonometric ratios of unknown angles.

Formula Used:
We will use the trigonometric identities tan2θ=sec2θ1 and secθ=1cosθ;

Complete step-by-step answer:
We will first solve the given equation 8tan2θ+9=6secθ.
By using the trigonometric identity tan2θ=sec2θ1, we get
8(sec2θ1)+9=6secθ
Multiplying the terms, we get
8sec2θ8+9=6secθ
Subtracting the like terms, we get
8sec2θ+1=6secθ
Rewriting the above equation, we get
8sec2θ6secθ+1=0
Above equation is a quadratic equation, so we will factorize the equation to find the value of θ.
Factorizing by grouping terms, we get
8sec2θ4secθ2secθ+1=0
Now factoring out the common term, we get
4secθ(2secθ1)1(2secθ1)=0

(4secθ1)(2secθ1)=0
Using zero product property, we get
(4secθ1)=04secθ=1secθ=14
Or
(2secθ1)=02secθ=1secθ=12
By using trigonometric identity secθ=1cosθ, we get
cosθ=4
Or
cosθ=2
We know that cosθ lies between 1<θ<1.
Since the values of cosθ does not lie between 1 and 1, so, there is no solution.
Therefore, the number of solutions of the equation 8tan2θ+9=6secθ in the interval (π2,π2) is zero.
Hence, the correct option is option C.

Note: We know that trigonometric equations are expressed as ratios of sine, cosine, tangent, cotangent, secant, cosecant angles. All possible values that satisfy the given trigonometric equation are called solutions of the given trigonometric equation. For a complete solution, “all possible values” satisfying the equation must be obtained. When we try to solve a trigonometric equation, we try to find out all sets of values of θ, which satisfy the given equation.