Question

(sin a + cos a )(sin a -cos a)/cos2a

Answer 1

$\frac{(Sin\alpha+Cos\alpha)(Sin\alpha-Cos\alpha)}{Cos2\alpha}=\frac{Sin^{2}\alpha-Cos^{2}\alpha}{Cos2\alpha}=\frac{-Cos2\alpha}{Cos2\alpha}=-1\\\\Otvet:\boxed{-1}$

Answer 2

To simplify the expression $(\sin a + \cos a)(\sin a - \cos a)/\cos^2 a$, let's break it down step by step:

Step 1: Distribute the multiplication $(\sin a + \cos a)(\sin a - \cos a) = \sin^2 a - \sin a \cos a + \sin a \cos a - \cos^2 a$

Step 2: Simplify the expression by canceling out like terms The middle terms $- \sin a \cos a + \sin a \cos a$ cancel each other out, so we're left with: $\sin^2 a - \cos^2 a$

Step 3: Apply the trigonometric identity $\sin^2 a - \cos^2 a = \sin^2 a - (1 - \sin^2 a)$ Since $\cos^2 a = 1 - \sin^2 a$

Simplifying further: $\sin^2 a - (1 - \sin^2 a) = \sin^2 a - 1 + \sin^2 a = 2\sin^2 a - 1$

Step 4: Divide by $\cos^2 a$ $\frac{2\sin^2 a - 1}{\cos^2 a}$

Step 5: Apply the trigonometric identity $\sin^2 a = 1 - \cos^2 a$ Since $\sin^2 a + \cos^2 a = 1$

Substituting the identity: $\frac{2(1 - \cos^2 a) - 1}{\cos^2 a} = \frac{2 - 2\cos^2 a - 1}{\cos^2 a} = \frac{1 - 2\cos^2 a}{\cos^2 a}$

This is the simplified form of the expression $(\sin a + \cos a)(\sin a - \cos a)/\cos^2 a$: $\frac{1 - 2\cos^2 a}{\cos^2 a}$