If $\sin(A) = \cos(A)$ find $2 \tan^2 A - 2 \sec^2 A + 5$

Question

This is my first try to trigonometry, I have solved 100s of questions but this one always comes incorrect!

Can anyone help me?

Question:

If $\sin A = \cos A$, find the value of:

$2 \tan^2 A - 2 \sec^2 A + 5$

Thanks a lot for any help!

Answer 1

Hint 1 : Since $\sin A$ and $\cos A$ are equal,

$$\tan A = \frac{\sin A}{\cos A} = 1$$

Hint 2 :

$$\sec^2 A = 1 + \tan^2 A$$

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In fact, for any real values of $A$, we have $\tan^2 A - \sec^2 A = -1$ by rearranging the Pythagorean identity.

As such, the required result is $-2 + 5 = 3$ regardless of any constraint on $A$.

Answer 2

$$2\tan^2(A)-2\sec^2(A)+5$$$$=2\frac{\sin^2(A)}{\cos^2(A)}-2\frac{1}{\cos^2(A)}+5$$$$=2\frac{\sin^2(A)-1}{\cos^2(A)}+5=-2+5=3$$