Question

Can someone please help me?

Verify each trigonometric equation by substituting identities to match the right hand side of the equation to the left hand side of the equation.

(sin x)(tan x cos x - cot x cos x) = 1 - 2 cos2x

1 + sec2x sin2x = sec2x

[(sin(x))/(1-cos(x))]+[(sin(x))/(1+cos(x))]=2csc(x)

- tan2x + sec2x = 1

Answer 1

I know I should start with the left side on the first one because it will be easier to manipulate.

Answer 2

I think (sin x)(tan x cos x- cot x cos x)= (SOMETHING)(sinx/cosxSOMETHING)

Answer 3

Good idea to start on the left.

Answer 4

(sin x)(tan x cos x - cot x cos x)
distribute
tanx cosx sinx - cotx cosx sinx
recall tanx = sinx / cosx and cotx = cosx/sinx

Answer 5

ok so then i have \[\frac{ \sin x }{ \cos x } \cos x \sin x - \frac{ \cos x }{ \sin x} \cos x \sin x\] thanks for replying btw

Answer 6

reduce to get
sin^2x - cos^2x

Answer 7

\[\frac{ \sin x }{ \cos x} \sin ^{2}x - \frac{ \cos x }{ \sin x} \cos ^{2}x\]

Answer 8

like that?

Answer 9

not quite...
\[ \large \frac{ \sin x }{ \color{red} {\cancel {\cos x} }} \color{red} {\cancel{\cos x}} \sin x - \frac{ \cos x }{ \color{red} {\cancel{\sin x}}} \cos x \color{red} {\cancel{\sin x}}\]

Answer 10

oh, ok. so then we have \[\sin ^{2}x-\cos ^{2}x\]

Answer 11

We are pretty close to the end... Just to verify befor going on...
Is the RHS
\[\large 1 - 2 \cos(2x)\]
or is it
\[\large 1 - 2\cos^2(x)\]

Answer 12

It's probably the second one.... I'm going to log off soon so I'll post the rest.

Use Double Angle Formulas
\[\large \cos(2x) = \cos^2x - \sin^2x = 2\cos^2x - 1\]
so
\[\large \sin^2x - \cos^2x \quad \color{green}{\text{factor out} -1}\]
\[\large = -1\left(\cos^2x - \sin^2x\right) \quad \color{green}{\text{use above identity}}\]
\[\large = -1\left(2\cos^2x - 1\right) \quad \color{green}{\text{distribute}}\]
\[\large = -2\cos^2x + 1 \quad \color{green}{\text{rewrite}}\]
\[\large = 1 -2\cos^2x \quad \color{green}{\text{done}}\]

Answer 13

OMG, I actually understand that. Thanks so much.

Answer 14

Totally makes sense!! I have been struggling for hours! I love you random citizen.

Answer 15

@cruffo