Prove the identity.
sin² θ = tan² θ / 1 + tan² θ
I assume the right side is supposed to be tan² è /(1 + tan² è)
Start by rewriting
1 + tan^2 = (cos^2 + sin^2)/cos^2
= 1 /cos^2
so that
1/(1+tan^2) = cos^2
Therefore the right side becomes
(sin^2/cos^2)* cos^2 = sin^2
how did 1 + tan^2 become cos^2 + sin ^2?
oh nevermind i got it thank you.
we know sec^2(x)-tan^2(x)=1
now sec^2(x)=1+tan^2(x)
now tan^2(x)/sec^2(x)=sin^2(x) [tan(x)=sin(x)/cos(x)]
[sec(x)=1/cos(x)]
To prove the given identity, we will manipulate the right-hand side of the equation and show that it simplifies to the left-hand side.
Start with the right-hand side of the equation:
tan²θ / (1 + tan²θ)
To simplify this expression, we need to use a trigonometric identity. One commonly used identity is:
tan²θ + 1 = sec²θ
Rearranging the identity, we get:
tan²θ = sec²θ - 1
Substituting this into the original expression:
tan²θ / (1 + tan²θ) = (sec²θ - 1) / (1 + sec²θ - 1)
Simplifying further:
(sec²θ - 1) / (sec²θ) = sec²θ / sec²θ
Using the identity sec²θ = 1 + tan²θ, we can rewrite the expression:
1 + tan²θ / sec²θ = 1 + tan²θ / (1 + tan²θ)
Since 1 + tan²θ / (1 + tan²θ) simplifies to 1, we have:
1 = 1
Therefore, we have proven the identity:
sin²θ = tan²θ / (1 + tan²θ)