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What is the simplest way to learn,solve, and prove trigonometric identities?  for example.....solve and prove the identity below: 1+(sec^2x)(sin^2x)= sec^2x

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lfryerda eNotes educator | Certified Educator

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To solve trigonometric identities, you often want to start on one side of the identity and need to somehow use simpler identities to get to the other side.

There is no fixed way of doing this, but there are usually several heuristics to follow.

 - convert everything to basic trig functions, such as sin, cos or tan - sometimes this will make things more complicated though.

 - be aware of more than just the identities `sin^2x+cos^2x=1` and `tanx = {sinx}/{cosx}`  - although not always used, they make your job easier when they are used.

Consider the identity presented.

`1+sec^2xsin^2x = sec^2x`   start with left side

`LS = 1+sec^2xsin^2x`   convert to definitions

`=1+{sin^2x}/{cos^2x}`   get common denominators

`={cos^2x+sin^2x}/{cos^2x}`   use identity in numerator

`=1/{cos^2x}`   use definition of secant

`=sec^2x`

`=RS`

The identity has been proven.

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