Prove the identity: tan^2O/ 1 + tan^20 = sin^20
I get 1=1, but others have got sin^2=sin^2
Who's right?? Because there is no answer for this question at the back of the book
To prove the identity tan^2O / (1 + tan^2O) = sin^2O, we can use trigonometric identities.
First, let's recall the identity for tan^2O:
tan^2O = sin^2O / cos^2O
Now, substitute this expression into the given equation:
(sin^2O / cos^2O) / (1 + (sin^2O / cos^2O))
To simplify this expression, we need to find a common denominator, which in this case is cos^2O:
[(sin^2O) / (cos^2O)] / [(cos^2O + sin^2O) / (cos^2O)]
Now, cancel out the cos^2O terms in the numerator and denominator:
sin^2O / (cos^2O + sin^2O)
Notice that the denominator cos^2O + sin^2O is equal to 1 according to the Pythagorean identity:
cos^2O + sin^2O = 1
So, the expression simplifies to:
sin^2O / 1
which is equal to sin^2O.
Hence, the identity tan^2O / (1 + tan^2O) = sin^2O is proven.
Regarding the discrepancy in the answers, it is important to remember that trigonometric identities are based on mathematical principles and should yield the same result. If your solution matches the above derivation, where sin^2O is obtained, it is correct. However, if there are different answers from others, it could be due to errors or different approaches in their calculations.