Question 130978
Let's manipulate the left side to transform it into the right side (note: we are not going to even touch the right side)



*[Tex \LARGE \tan^2\left(x\right)-\sin^2\left(x\right)]  .... Start with the left side



*[Tex \LARGE =\frac{\sin^2\left(x\right)}{\cos^2\left(x\right)}-\sin^2\left(x\right)]  .... Rewrite *[Tex \Large \tan^2\left(x\right)] as *[Tex \Large \frac{\sin^2\left(x\right)}{\cos^2\left(x\right)}]



*[Tex \LARGE =\frac{\sin^2\left(x\right)}{\cos^2\left(x\right)}-\sin^2\left(x\right)\left(\frac{\cos^2\left(x\right)}{\cos^2\left(x\right)\right)]   .... Multiply *[Tex \Large \sin^2\left(x\right)] by *[Tex \Large \frac{\cos^2\left(x\right)}{\cos^2\left(x\right)}]



*[Tex \LARGE =\frac{\sin^2\left(x\right)}{\cos^2\left(x\right)}-\frac{\sin^2\left(x\right)\cos^2\left(x\right)}{\cos^2\left(x\right)}]   .... Multiply 



*[Tex \LARGE =\frac{\sin^2\left(x\right)-\sin^2\left(x\right)\cos^2\left(x\right)}{\cos^2\left(x\right)}]   .... Combine the fractions. 



*[Tex \LARGE =\frac{\sin^2\left(x\right)\left(1-\cos^2\left(x\right)\right)}{\cos^2\left(x\right)}]   .... Factor out *[Tex \Large \sin^2\left(x\right)]



*[Tex \LARGE =\frac{\sin^2\left(x\right)\left(sin^2\left(x\right)\right)}{\cos^2\left(x\right)}]   .... Replace *[Tex \Large 1-\cos^2\left(x\right)] with *[Tex \Large \sin^2\left(x\right)]



*[Tex \LARGE =\tan^2\left(x\right)\sin^2\left(x\right)]   .... Replace *[Tex \Large \frac{\sin^2\left(x\right)}{\cos^2\left(x\right)}]  with *[Tex \Large \tan^2\left(x\right)]





So this shows us that



*[Tex \LARGE \tan^2\left(x\right)-\sin^2\left(x\right)=\tan^2\left(x\right)\sin^2\left(x\right)]