Thread: trigonometric

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

Originally Posted by xavior

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

1. 5 + sin^2 A=sinA + 5 ----------------------------(1)
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. ---------*****

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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.

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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.

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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