Question

prove trig identity:

1 + (sec^2 x)(sin^2x) = sec^2x

Answer 1

\[1+(\sec^2x)(\sin ^{2}x)=\sec ^{2}x\] you can rewrite \[\sec ^{2}x = \frac{ 1 }{ \cos ^{2}x }\]so you have \[1+\frac{ \sin ^{2}x }{ \cos ^{2}x }=\sec ^{2}x\]we know that \[\frac{ \sin ^{2}x }{ \cos ^{2}x}=\tan ^{2}\] so inputting that into our equation we have \[1 + \tan ^{2}x = \sec ^{2}x\] So \[\sec ^{2}x = \sec ^{2}x\]

Answer 2

sorry, i meant \[\tan ^{2}x \]made a typo.

Answer 3

to be a bit more thorough .
\[1 + \frac{\sin^{2} x}{\cos^{2} x} = \frac{\cos^{2} x}{\cos^{2} x} +\frac{\sin^{2} x}{\cos^{2} x} = \frac{\sin^{2} x+\cos^{2} x}{\cos^{2} x} = \frac{1}{\cos^{2} x} = \sec^{2} x\]

Answer 4

And just remember when you write a proof on paper to put \(\implies\) as the start of each new line and only one operation per line. The joy of proofs... and how they waste paper!

Answer 5

IKR

Answer 6

Oh, and the Q.E.D. In Linear Algebra I had a prof that would ding you if you forgot the // or Q.E.D. or \(\blacksquare\) at the end.