Prove the identity.
2tanx/1+tan^2x=sin^2x
Not so.
if x=pi/4,
2*1/(1+1) = 1
sin^2 pi/4 = 1/2
I think you mean
2tanx/(1+tan^2x)=sin2x
since
2sinx/cosx / sec^2x
2sinx/cosx * cos^2x
2sinx*cosx = sin2x
If I garbled it, make the corrections and repost if needed.
To prove the given identity 2tanx / (1 + tan^2x) = sin^2x, we will manipulate the left-hand side (LHS) of the equation until it matches the right-hand side (RHS). Here's how:
Starting with LHS:
2tanx / (1 + tan^2x)
Step 1: Rewrite tan^2x as sin^2x / cos^2x using the Pythagorean identity, tan^2x + 1 = sec^2x.
2tanx / (1 + sin^2x / cos^2x)
Step 2: To simplify the expression further, multiply the numerator and denominator by cos^2x to eliminate the fractions:
2tanx * cos^2x / (cos^2x + sin^2x)
Step 3: Recall that cos^2x + sin^2x = 1 due to the Pythagorean identity for both sine and cosine.
2tanx * cos^2x / 1
Simplifying further:
2tanx * cos^2x
Step 4: Use the identity tanx = sinx / cosx to express tanx in terms of sine and cosine.
2(sin x / cos x) * cos^2x
Step 5: Multiply 2 with cos^2x:
2sin x * cos x / cos x
Step 6: Cancel out a common factor of cos x:
2sin x
The simplified expression matches the RHS:
sin^2x
Therefore, we have proven that 2tanx / (1 + tan^2x) = sin^2x.