Question

Prove this identity. Tan^2(0)/1 + tan^2(0) = sin^2(0). Note i placed a bracket around the zeros because i didnt want to look like it is to the power of 20. And the zero is supposed to be the symbol theta.

Answer 1

Use the trigonometric identities:

`\tan \theta = (\sin \theta)/(\cos \theta) \\ \sin^2 \theta+ \cos^2 \theta=1`

Proof:

`(\tan^2 \theta)/(1+ \tan^2 \theta)}=\sin^2 \theta \ \ \ |* (1+\tan^2 \theta) \\ \\ \tan^2 \theta = \sin^2 \theta (1+ \tan^2 \theta) \ \ \ |\hbox{convert } \tan \theta \hbox{ to } (\sin \theta)/(\cos \theta) \\ \\ ((\sin \theta)/(\cos \theta))^2=\sin^2 \theta (1+ ((\sin \theta)/(\cos \theta))^2) \\ \\ (\sin^2 \theta)/(\cos^2 \theta)=\sin^2 \theta(1+(\sin^2 \theta)/(\cos^2 \theta)) \ \ \ |/ \sin \theta`


`(1)/(\cos^2 \theta)=1+(sin^2 \theta)/(\cos^2 \theta) \ \ \ |* \cos^2 \theta \\ \\ 1=\cos^2 \theta+\sin^2 \theta \ \ \ |\hbox{convert } \cos^2 \theta+ \sin^2 \theta \hbox{ to } 1 \\ \\ 1=1`