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Sati [7]
4 years ago
14

Which shows all the exact solutions of 2sec^2x-tan^4x=-1 ? Give your answer in radians.

Mathematics
2 answers:
Kazeer [188]4 years ago
7 0

the answer is A on edg

dangina [55]4 years ago
4 0
Lets solve the trigonometric equation 2sec^2(x)-tan^4(x)=1

Step 1. Use the trigonometric identity: sec^2(x)=1+tan^2(x):
2[1+tan^2(x)]-tan^4(x)=1
2+2tan^2(x)-tan^4(x)=1

Step 2. Subtract 1 from both sides of the equation:
2-1+2tan^2(x)-tan^4(x)=1-1
1+2tan^2(x)-tan^4(x)=0

Step 3. Substitute tan^2(x)=u:
1+2u-u^2=0

Step 3. Multiply both sides of the equation by -1:
-1(1+2u-u^2)=0(-1)
u^2-2-1=0

Step 4. Solve for u:
u=1+ \sqrt{2} ,u=1- \sqrt{2}

Step 5. Substitute tan^2(x)=u:
tan^2(x)=1+ \sqrt{2} ,tan^2(x)=1- \sqrt{2}

Step 6. Solve for x:
\sqrt{tan^2(x)} =(+/-) \sqrt{1+ \sqrt{2}} , \sqrt{tan^2(x)}= (+/-)\sqrt{1- \sqrt{2} }
tan(x)=\sqrt{1+ \sqrt{2}},-\sqrt{1+ \sqrt{2}},\sqrt{1- \sqrt{2} },-\sqrt{1- \sqrt{2} }
Since \sqrt{1- \sqrt{2} } is not a real root, we can rule out \sqrt{1- \sqrt{2} },-\sqrt{1- \sqrt{2} }
tan(x)=\sqrt{1+ \sqrt{2}},-\sqrt{1+ \sqrt{2}}
x=arctan(\sqrt{1+ \sqrt{2}})+k \pi ,x=arctan(-\sqrt{1+ \sqrt{2}})+k \pi
x=0.9989+k \pi ,x=-0.9989+k \pi

We can conclude that the exact solution of the equation are x=arctan(\sqrt{1+ \sqrt{2}})+k \pi and x=arctan(-\sqrt{1+ \sqrt{2}})+k \pi, which are approximately x=0.9989+k \pi and x=-0.9989+k \pi respectively. 

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