ANSWER
k = 1 or 21
STEP-BY-STEP EXPLANATION:
According to the question, we were given the below trigonometric function
[tex]\sec ^2x\text{ - 22tanx + 20 = 0}[/tex]
Recall that, we have trigonometric identity which is written below
[tex]\sin ^2\theta+cos^2\theta\text{ = 1}[/tex]
[tex]\text{Divide through by }\cos ^2\theta[/tex][tex]\begin{gathered} \frac{\sin^2\theta}{\cos^2\theta}\text{ + }\frac{cos^2\theta}{\cos^2\theta}\text{ =}\frac{1}{\cos ^2\theta} \\ \tan ^2\theta+1=sec^2\theta \\ \text{Let x = }\theta \\ \tan ^2x+1=sec^2x \end{gathered}[/tex]
The next thing is to rewrite the equation
[tex]\begin{gathered} \text{ since sec}^2x=tan^2x\text{ + 1} \\ \text{Hence,} \\ \tan ^2x\text{ + 1 - 22tanx + 20 = 0} \\ \text{Let k = tanx} \\ k^2\text{ + 1 -22k + 20 = 0} \\ \text{Collect the like terms} \\ k^2\text{ - 22k + 21 = 0} \end{gathered}[/tex]
The next thing is to find the value of P by factorizing the above equation.
Recall that, the standard form of the quadratic function is given as
[tex]ax^2\text{ + bx + c = 0}[/tex]
Let
a = 1
b = -22
c = 21
The next thing is to find the value of ac
[tex]\begin{gathered} ac\text{ = 1 }\cdot\text{ 2}1 \\ ac\text{ = 2}1 \end{gathered}[/tex][tex]\begin{gathered} k^2\text{ - k -21k + 21 =0} \\ k(k\text{ -1) -21(}k\text{- 1) = 0} \\ (k\text{ -1) (k -21) = 0} \\ k\text{ -1 = 0 or =k - 21 = 0} \\ k\text{ = 1 or k = 22} \end{gathered}[/tex]
Hence, the value of k is 1 or 21
