How Do You Solve sin^2(x) + tan^2(x) = √2?

[ForceBoy]

Homework Statement

The problem given is sin²(x) + tan²(x) = √2

2. Homework Equations

The relevant equations would be any trigonometric identities

The Attempt at a Solution

sin²(x) + tan²(x) = √2

sin²(x) + (sin²(x)/cos²(x) ) = √2

[ cos²(x) sin²(x) + sin²(x) ]/ cos²(x) = √2

[ (1- sin²(x)) sin²(x) + sin²(x)] / [ 1 - sin²(x) ] = √2

[ (2 - sin²(x) ) sin²(x) ] / [ 1 - sin²(x) ] = √2

(2 - sin²(x) ) tan²(x) = √2

tan²(x) = ( √2 / [ 2 - sin²(x) ] )

I take this and substitute into the first equation:

sin²(x) + ( √2 / [ 2 - sin²(x) ] ) = √2

( 2 - sin²(x) ) sin²(x) / [ 2 - sin²(x) ] + √2 / [ 2 - sin²(x) ] = √2

( 2 - sin²(x) ) sin²(x) + √2 / [ 2 - sin²(x) ] = √2

( 2 - sin²(x) ) sin²(x) + √2 = ( √2 ) 2 - sin²(x)

( 2 - sin²(x) ) sin²(x) + √2 = 2√2 - √2 sin²(x)

( 2 - sin²(x) ) sin²(x) = √2 - √2 sin²(x)

( 2 - sin²(x) ) sin²(x) = √2 (1 - sin²(x) )

( ( 2 - sin²(x) ) sin²(x) / (1 - sin²(x) ) )= √2Here is where I get stuck. I do not know what steps to take next. Please give me hints on this and do not hesitate to point out any mistakes in my work. They are very likely.

 

[fresh_42]

Homework Statement

The problem given is sin²(x) + tan²(x) = √2

2. Homework Equations

The relevant equations would be any trigonometric identities

The Attempt at a Solution

sin²(x) + tan²(x) = √2

sin²(x) + (sin²(x)/cos²(x) ) = √2

[ cos²(x) sin²(x) + sin²(x) ]/ cos²(x) = √2

[ (1- sin²(x)) sin²(x) + sin²(x)] / [ 1 - sin²(x) ] = √2

[ (2 - sin²(x) ) sin²(x) ] / [ 1 - sin²(x) ] = √2

I can follow you until here. What's next is a backward substitution which I don't think will get you very far. At least the next steps are what could be done more easily, because you already have a quadratic equation in $t := \sin^2(x)$ which can be solved.

(2 - sin²(x) ) tan²(x) = √2

tan²(x) = ( √2 / [ 2 - sin²(x) ] )

I take this and substitute into the first equation:

sin²(x) + ( √2 / [ 2 - sin²(x) ] ) = √2

( 2 - sin²(x) ) sin²(x) / [ 2 - sin²(x) ] + √2 / [ 2 - sin²(x) ] = √2

( 2 - sin²(x) ) sin²(x) + √2 / [ 2 - sin²(x) ] = √2

( 2 - sin²(x) ) sin²(x) + √2 = ( √2 ) 2 - sin²(x)

( 2 - sin²(x) ) sin²(x) + √2 = 2√2 - √2 sin²(x)

( 2 - sin²(x) ) sin²(x) = √2 - √2 sin²(x)

( 2 - sin²(x) ) sin²(x) = √2 (1 - sin²(x) )

( ( 2 - sin²(x) ) sin²(x) / (1 - sin²(x) ) )= √2Here is where I get stuck. I do not know what steps to take next. Please give me hints on this and do not hesitate to point out any mistakes in my work. They are very likely.

 

[ForceBoy]

Wow, I didn't see that! Thank you very much. This was really helpful.