Question

Simplify the expression:
Sin^2x-1/cos(-x)

Answer 1

Have you seen the following identity?

\(\large\color{black}{ \displaystyle \sin^2\theta+\cos^2\theta=1 }\)

Answer 2

Yes

Answer 3

Subtract \(\sin^2\theta\) from both sides and tell me what you get.

Answer 4

\(\large\color{black}{ \displaystyle \sin^2\theta+\cos^2\theta =1 \ }\)

\(\large\color{black}{ \displaystyle \sin^2\theta+\cos^2\theta \color{blue}{-\sin^2\theta }=1 \color{blue}{-\sin^2\theta } }\)

\(\large\color{black}{ \displaystyle \cos^2\theta =1-\sin^2\theta }\)

Answer 5

So, based on the very last statement, you can now simplify your expression:

\(\large\color{black}{ \displaystyle \frac{1-\sin^2x}{\cos x} =\frac{?}{\cos x} }\)

Answer 6

hello?

Answer 7

Sinx?

Answer 8

We have concluded that:
\(1-\sin^2\theta =\cos^2\theta\)

correct?

Answer 9

Yes so the ? Represents cos^2 theta?

Answer 10

Yes, therefore it comes out that:

\(\large\color{black}{ \displaystyle \frac{1-\sin^2x}{\cos x} =\frac{\cos^2x}{\cos x} }\)

Answer 11

and THAT \(\Uparrow\) you can probably tell me how to simplify:D

Answer 12

So the answer is cos x? :)

Answer 13

yes, just \(\cos x\).

Answer 14

Any questions?

Answer 15

Thank you for your help! i understand it now :D

Answer 16

Yes, one more note:
\(\color{red}{\cos(-\theta)=\cos(\theta)}\)

Answer 17

So, lets quicly recap everything once more, okay?

Answer 18

\(\large\color{black}{ \displaystyle \frac{1-\sin^2(x)}{\cos (-x)} = }\) \(\large\color{black}{ \displaystyle \frac{1-\sin^2(x)}{\cos (x)} = }\) \(\large\color{black}{ \displaystyle \frac{\cos^2(x)}{\cos (x)} = \cos(x) }\)