[tex]3tan^{2} \theta +7sec\theta=3[/tex]
First I converted the equation terms into sine and cosine.
[tex]tan^{2}\theta = \frac{sin^{2}\theta}{cos^{2}\theta} [/tex] and [tex]sec\theta= \frac{1}{cos\theta} [/tex]
Substitution:
[tex] \frac{3sin^2\theta}{cos^2\theta} + \frac{7}{cos\theta} =3[/tex]
Common Denominator Created:
[tex] \frac{3sin^2\theta}{cos^2\theta} + \frac{7cos\theta}{cos^2\theta} =3[/tex]
Multiply each term by the LCD:
[tex]3sin^2\theta+7cos\theta=3cos^2\theta[/tex]
Substitution: Recall ⇒[tex]sin^2\theta =1-cos^2\theta[/tex]
[tex]3(1-cos^2\theta)+7cos\theta=3cos^2\theta[/tex]
Distribute and collect all terms on one side:
[tex]6cos^2\theta-7cos\theta-3=0[/tex]
Factor and set each factor equal to 0:
[tex](2cos\theta-3)(3cos\theta+1)=0[/tex]
[tex]2cos\theta-3=0[/tex]⇒[tex]theta=cos^{-1} \frac{3}{2} [/tex]
[tex]3cos\theta+1=0[/tex]⇒[tex]theta=cos^{-1} \frac{-1}{3} [/tex]
The 2nd factor provides only possible answer 109.5 degrees
