Question

Find a numerical value of one trigonometric function of x if $\frac{tan^{2} x-sin^{2}x}{sin^{2} x} = 5$

Answer 1

(tan²(x) - sin²(x)) / sin²(x) = 5

tan²(x) / sin²(x) - sin²(x) / sin²(x) = 5

(sin²(x) / cos²(x)) / sin²(x) - sin²(x) / sin²(x) = 5

If sin(x) ≠ 0, then the equation reduces to

1 / cos²(x) - 1 = 5

sec²(x) = 6

sec(x) = ± √6

x = arcsec(√6) + 2nπ   or   x = arcsec(-√6) + 2nπ

(where n is any integer)