Posted by **Evelina** on Wednesday, November 30, 2011 at 8:09am.

I need to prove equality.

a) (sina + cosa)^2 -1 / ctga - sinacosa = 2tg^2a

b) (sin^2x/sinx-cosx) - (sinx+cosx/tg^2x+1) = sinx + cosx

c) sin^4a - sin^2a - cos^4a + cos^2a = cosπ/2

How to do these?

- trig -
**Reiny**, Wednesday, November 30, 2011 at 8:28am
a)

LS = [sin^2a + 2sinacosa + cos^2a - 1] / [ cosa/sina - sinacosa]

= [1 + 2sinacosa -1] / [ (cosa - sin^2acosa)/sina]

= 2sinacosa/[ cosa(1 - sin^2a)/sina]

= 2sinacosa(sina/(cosa(cos^2a))

= 2 sin^2a/cos^2a

= 2tan^2a

= RS

b) The way you typed it, LS ≠ RS

Did you mean

(sin^2x/(sinx-cosx)) - ((sinx+cosx)/tg^2x+1) = sinx + cosx ?

Please clarify

c) LS = sin^2(sin^2a - 1) - cos^2a(cos^2a -1)

= sin^2a(-cos^2a) - cos^2a(-sin^2a)

= 0

RS = cos π/2

= 0

= LS

- trig -
**Evelina**, Wednesday, November 30, 2011 at 8:54am
The b) actually was like this in my book: sin^2x/sinx-cosx - sinx+cosx/tg^2x+1 = sinx + cosx

- trig -
**Reiny**, Wednesday, November 30, 2011 at 9:23am
When I can't seem to get anywhere with an identity I take any value of the variable and test it in the equation.

I tried x = 20° in

sin^2x/sinx-cosx - sinx+cosx/tg^2x+1 = sinx + cosx and LS ≠ RS

I tried it in

sin^2x/(sinx-cosx) - (sinx+cosx)/(tg^2x+1) = sinx + cosx and LS ≠ RS

I tried it in

sin^2x/(sinx-cosx) - (sinx+cosx)/tg^2x+1 = sinx + cosx and LS ≠ RS

You do realize that you must put brackets in this way of typing to identify which is the numerator and which is the denominator.

the way you typed it, the LS would have 5 terms

[sin^2x/sinx] - [cosx] - [sinx] + [cosx/tan^2x] + 1

I am pretty sure that is not what the question says.

I have a feeling there are two fractions

numerator of 1st fraction : sin^2x

denominator of 1st fraction: sinx - cosx

numerator of 2nd : sinx + cosx

denom of 2nd : tan^2x + 1

Thus

sin^2x/(sinx-cosx) - (sinx + cosx)/(tan^2x + 1) = sinx + cosx

and if I test x=20°, LS ≠ RS

- trig -
**Evelina**, Wednesday, November 30, 2011 at 9:34am
Well, that‘s what the question says. I can send a picture of it, if you don‘t believe.

Show me how you do it with brackets, maybe I‘ll know what to do with that one.

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