Simplifying trigonometric identities?

Question

I just can't figure out:

1. sin²x + tan²x + cos²x

and

2. cot(π/2 -x)cosx

Thanks in advance!

Answer 1

Remember sin^2(x) + cos^2(x) = 1.  So apply that identity to get

1 + tan^2(x).

Another identity (which you can derive from the one just mentioned) is tan^2(x) + 1 = sec^2(x).  So the answer simplifies to sec^2(x).

For the second one, cot(π/2 -x) = cos(π/2 -x) / sin(π/2 -x) =

sin(x) / cos(x).  Multiplying this by cos(x) gives sin(x).

Answer 2

1. sin²x + tan²x + cos²x

First of all... sin^2(x) + cos^2(x) = 1

So... sin²x + tan²x + cos²x = 1 + tan^2(x)

And that is a Pythagorean identity

1 + tan^2(x) = sec^2(x)  ANSWER.

2. cot(π/2 -x)cosx

cot(pi/2 - x) = tan(x)

So... cot(π/2 -x)cosx = tan(x)cos(x) = (sin(x) / cos(x)) cos(x) = sin(x)

And that's your answer.

Take care,

David

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