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gavmur [86]
3 years ago
12

Please help with this problem.

Mathematics
1 answer:
larisa86 [58]3 years ago
6 0

Answer:

60°, 120°

Step-by-step 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 = \sqrt{ - 1}  \: which \: is \: not \: possible \\  \therefore \: {cos}^{2}x =  \frac{1}{4}  \\  \\ \therefore \: {cos}x =   \pm\frac{1}{2} \\  \\ \therefore \: {cos}x =   \frac{1}{2}  \: or \: {cos}x =    - \frac{1}{2}  \\  \\ \therefore \: {cos}x =   {cos}60 \degree \: or \: {cos}x =     {cos}120 \degree \\  \\ \therefore \:x = 60 \degree \:  \: or  \: \: x  = 120 \degree

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while if k is even, then the sum would be

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