Help simplifying (secθ + cscθ)(cosθ-sinθ)?

Question

Equation: (secθ + cscθ)(cosθ-sinθ)

So far, I have
( (1/cosθ) + (1/sinθ) ) * ( (cosθ/1) - (sinθ/1) )

Then I distributed and got ( (2cosθ-2sinθ) / 0 ) and got an error. Where did I go wrong? Thanks for the help.

Can you simplify (cos^2θ * sin^2θ) into anything smaller or is that the simplest form? Thanks.

Answer 1

A.) Your first EXPRESSION (not "equation") simplifies to
2 cot (2θ).

B.) The expression in your "Additional details" (which are NOT additional details, but rather IS a completely separate, additional question) simplifies to (1/4) sin^2 (2θ).

These separate simplifications are done as follows:

A.) : (sec θ + csc θ) (cos θ - sin θ)

= (1 / cos θ + 1 / sin θ) (cos θ - sin θ)

= 1 + cos θ / sin θ - sin θ / cos θ - 1

= (cos^2 θ - sin^2 θ) / (sin θ cos θ)

= cos (2θ) * 2 / sin (2θ) = 2 cot (2θ).

B.) : (cos^2 θ * sin^2 θ) = (sin θ * cos θ)^2 = (1/4) sin^2 (2θ).

QED

Live long and prosper.

Answer 2

(secθ + cscθ) (cosθ - sinθ)

Changing everything to sines and cosines,

(1/cosθ + 1/sinθ) (cosθ - sinθ)

Using FOIL,

cosθ/cosθ - sinθ/cosθ + cosθ/sinθ - sinθ/sinθ

1 - sinθ/cosθ + cosθ/sinθ - 1

Note the cancellation.

-sinθ/cosθ + cosθ/sinθ

Now, putting this under a common denominator of sin(θ)cos(θ), we get

[-sin^2(θ) + cos^2(θ)] / [sin(θ)cos(θ)]

Rearranging the top,

[cos^2(θ) - sin^2(θ)] / [sin(θ)cos(θ)]

This is the part that requires intuition. By the double angle identity

cos(2θ) = cos^2(θ) - sin^2(θ)

sin(2θ) = 2sin(θ)cos(θ). For this equation, multiply both sides by (1/2), we get

(1/2)sin(2θ) = sin(θ)cos(θ)

Therefore,

[cos^2(θ) - sin^2(θ)] / [sin(θ)cos(θ)]

becomes

cos(2θ) / [ (1/2) sin(2θ) ]

Multiply top and bottom by 2,

2 cos(2θ)/sin(2θ)

Which, by definition, is equal to

2cot(2θ)

Side note: To avoid confusion in the future, remember to use the word "expression" instead of "equation" if there's no equal sign involved. I was initially confused when you declare an equation, but I could not find an equation.

Answer 3

(sec x + cosec x)(cos x - sin x)
= (1/cos x + 1/sin x)(cos x - sinx)
= (sin x + cos x)(cos x - sin x)/(cos x sin x)
= (cos^2 x - sin^2 x)/cos x sin x
= 2(cos 2x) / sin 2x
= 2cot(2x)
I'm afraid I don't understand how you arrived at your division by 0.

Answer 4

sorry i replaced the attitude theta with the help of x,yet you recognize that no longer something replaced . csc x=a million/sinx and sec x=a million/cos x then (sinx/csc x)+(cos x/sec x)=(sinx/(a million/sin x))+(cosx(/a million/cosx))=sin^2x+cos^2x= a million

Answer 5

(cos^2x * sin^2x) = cos^2xsin^2x = (sinxcosx)^2 = 1/4 (sin2x)^2

Answer 6

((1/cos t)+(1/sin t))(cos t) - ((1/cos t)+(1/sin t))(sin t)
1+(cos t)/(sin t)-((sin t)/(cos t)+1)
1+(cot t)-(tan t)-1
cot t - tan t