To confirm

1. 5 + sin^2 A=sinA + 5

2. cos^4 A-sin^4 A=cos2A

3. cos^2 A (2tan^2 A + sec^2 A)= 2-cos2A

Printable View

- March 2nd 2006, 12:56 AMxaviortrigonometric
To confirm

1. 5 + sin^2 A=sinA + 5

2. cos^4 A-sin^4 A=cos2A

3. cos^2 A (2tan^2 A + sec^2 A)= 2-cos2A - March 2nd 2006, 02:40 AMticbolQuote:

Originally Posted by**xavior**

When in doubt, put a numerical value to the angle.

Say angle A = 30deg

5 +sin^2(30deg) =? sin(30deg) +5

5 +(0.5)^2 =? 0.5 +5

5.25 =? 5.5

No.

So (1) is not true for A=30deg. Therefore, (1) is not confirmed for all values of A.

Meaning, (1) is not an identity.

An identity is true for all the values of the angle in question. ---------*****

---------------------------------------------------------

2. cos^4 A-sin^4 A = cos2A ------------(2)

I tried A=30deg and it proved true, so, maybe (2) is an identity.

Let us see.

Lefthand Side, (LHS) =

= cos^4(A) -sin^4(A)

Using a^2 -b^2 = (a+b)(a-b),

= [cos^2(A) +sin^2(A)]*[cos^2(A) -sin^2(A)]

= [1]*[cos(2A)]

= cos(2A)

= Righthand Side, RHS.

Therefore, confirmed.

-----------------------------------------------------

3. cos^2 A (2tan^2 A + sec^2 A)= 2-cos2A

I tried again A=30deg and (3) was true, so, maybe (3) is an identity.

LHS =

= cos^2(A) *[2tan^2(A) +sec^2(A)]

= 2cos^2(A)*tan^2(A) +cos^2(A)*sec^2(A)

= 2cos^2(A)[sin2(A)/cos^2(A)] +cos^2(A)[1/cos^2(A)]

= 2sin^2(A) +1

= 2[1 -cos^2(A)] +1

= 2 -2cos^2(A) +1

= 2 -2cos^2(A) +[sin^2(A) +cos^2(A)]

= 2 -cos^2(A) +sin^2(A)

= 2 -[cos^2(A) -sin^2(A)]

= 2 -[cos(2A)]

= 2 -cos(2A)

= RHS

Therefore, confirmed.

--------------------------------------

We used the trig identities:

---sin^2(A) +cos^2(A) = 1

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

---tanA = sinA/cosA

---secA = 1/cosA