What can you do to the equation sin^2(θ)+2θ+cos^2(θ)=1 to get the identity 1+cot^2(θ)=csc 2^(θ)?
There may be more than one correct answer. Select all that apply.
Rewrite sin^2(θ)/cos^2(θ) using the tangent identity.
Simplify cos^2(θ)/cos^2(θ) to get 1.
Rewrite 1/cos^2(θ) using the reciprocal identity for secant.
Simplify sin^2(θ)sin^2(θ) to get 1.
Divide both sides of the equation by cos^2(θ).
Rewrite 1/sin^2(θ) using the reciprocal identity for cosecant.
Divide both sides of the equation by sin^2(θ).
Rewrite cos^2(θ)/sin^2(θ) using the cotangent identity.
If sin^2(θ)+2θ+cos^2(θ)=1 means:
sin² ( θ ) + cos² ( θ ) = 1
then
sin² ( θ ) + cos² ( θ ) = 1
Divide both sides by sin² ( θ )
sin² ( θ ) / sin² ( θ ) + cos² ( θ ) / sin² ( θ ) = 1 / sin² ( θ )
1 + [ cos ( θ ) / sin ( θ ) ]² = [ 1 / sin ( θ ) ]²
1 + cot² ( θ ) = csc² ( θ )
where did that 1 come from in step 3?
nevermind, that is sin^2(θ )/sin^2(θ ), Thank you!!!!
To get the identity 1+cot^2(θ)=csc^2(2θ) from the given equation sin^2(θ)+2θ+cos^2(θ)=1, we can follow these steps:
1. Rewrite sin^2(θ)/cos^2(θ) using the tangent identity:
tan^2(θ) + 2θ + cos^2(θ) = 1
2. Simplify cos^2(θ)/cos^2(θ) to get 1:
tan^2(θ) + 2θ + 1 = 1
3. Rewrite 1/cos^2(θ) using the reciprocal identity for secant:
tan^2(θ) + 2θ + sec^2(θ) = 1
4. Simplify sin^2(θ)sin^2(θ) to get 1:
tan^2(θ) + 2θ + sec^2(θ) = sin^2(θ)
5. Divide both sides of the equation by cos^2(θ):
tan^2(θ)/cos^2(θ) + 2θ/cos^2(θ) + sec^2(θ)/cos^2(θ) = sin^2(θ)/cos^2(θ)
6. Rewrite 1/sin^2(θ) using the reciprocal identity for cosecant:
cot^2(θ) + 2θ/cos^2(θ) + sec^2(θ)/cos^2(θ) = csc^2(θ)
7. Divide both sides of the equation by sin^2(θ):
cot^2(θ)/sin^2(θ) + 2θ/cos^2(θ)sin^2(θ) + sec^2(θ)/cos^2(θ)sin^2(θ) = csc^2(θ)/sin^2(θ)
8. Rewrite cos^2(θ)/sin^2(θ) using the cotangent identity:
cot^2(θ) + 2θcot^2(θ) + sec^2(θ)csc^2(θ) = csc^2(θ)/sin^2(θ)
Therefore, we can see that the correct steps to obtain the identity 1+cot^2(θ)=csc^2(2θ) from the given equation are:
- Rewrite sin^2(θ)/cos^2(θ) using the tangent identity.
- Simplify cos^2(θ)/cos^2(θ) to get 1.
- Rewrite 1/cos^2(θ) using the reciprocal identity for secant.
- Simplify sin^2(θ)sin^2(θ) to get 1.
- Divide both sides of the equation by cos^2(θ).
- Rewrite 1/sin^2(θ) using the reciprocal identity for cosecant.
- Divide both sides of the equation by sin^2(θ).
- Rewrite cos^2(θ)/sin^2(θ) using the cotangent identity.