Question

Please help with this problem.

Answer 1

60°, 120°

Explanation:

`\frac{ {tan}^(2)x }{2} - 2 {cos}^(2)x = 1 \\ \\ \frac{ {tan}^(2)x - 4{cos}^(2)x }{2} = 1 \\ \\ {tan}^(2)x - 4{cos}^(2)x = 2 \\ \\ \frac{{sin}^(2)x}{{cos}^(2)x} - 4{cos}^(2)x = 2 \\ \\ \frac{{sin}^(2)x - 4{cos}^(4)x}{{cos}^(2)x} = 2 \\ \\ {sin}^(2)x - 4{cos}^(4)x = 2{cos}^(2)x \\ \\ 4{cos}^(4)x + 2{cos}^(2)x - {sin}^(2)x = 0 \\ \\ 4{cos}^(4)x + 2{cos}^(2)x + {cos}^(2)x - 1= 0 \\ \\ 4{cos}^(4)x + 3{cos}^(2)x - 1= 0 \\ \\ 4{cos}^(4)x + 4{cos}^(2)x - {cos}^(2)x - 1= 0 \\ \\4{cos}^(2)x({cos}^(2)x + 1) - 1({cos}^(2)x + 1) = 0 \\ \\ ({cos}^(2)x + 1)(4{cos}^(2)x - 1) = 0 \\ \\ ({cos}^(2)x + 1) = 0 \: or \: (4{cos}^(2)x - 1) = 0 \\ \\ {cos}^(2)x = - 1 \: or \: 4{cos}^(2)x = 1 \\ \\ {cos}x = √( - 1) \: which \: is \: not \: possible \\ \therefore \: {cos}^(2)x = (1)/(4) \\ \\ \therefore \: {cos}x = \pm(1)/(2) \\ \\ \therefore \: {cos}x = (1)/(2) \: or \: {cos}x = - (1)/(2) \\ \\ \therefore \: {cos}x = {cos}60 \degree \: or \: {cos}x = {cos}120 \degree \\ \\ \therefore \:x = 60 \degree \: \: or \: \: x = 120 \degree`