Perform the operation and simplify

Question

Sin x cox (tan x +cot x)

Answer 1

Simplify sin x cos x (tan x + cot x)

 

To simplify, you need to know a few trig identities.  The ones you'll need in this problem are:

 

1)  tan x = sin x/cos x

2)  cot x = cos x/sin x

3)  sin² x + cos² x = 1

 

If you substitute the first two identities (#1 and 2) into your expression:

 

sin x cos x (tan x + cot x) = sin x cos x (sin x/cos x + cos x/sin x)

 

Since the answer cat is now out of the bag, I'll finish the solution for you.  Add the two terms in the parentheses by putting them over a common denominator (like you would do to add any fractions), then add them.  This will give you:

 

sin x cos x [ (sin²x+cos²x)/sin x cos x ]

 

Since sin²x + cos²x = 1 (by identity #3 above), the expression becomes:

 

sin x cos x / sin x cos x = 1

 

I hope that helps

Answer 2

sin(θ)*cox(θ)*(tan(θ) + cot(θ)) = ?

 

Graph a general point P(x,y) in Quadrant I (works in any Quadrant; just more intuitive in I).

 

Draw a segment from A(0,0) to P(x,y).

 

Draw a segment from P(x,y) to B(x,0).

 

Label the segment lengths: AB = x, BP = y, AP = r

 

By the Pythagorean Theorem r = √(x^2+y^2), where r is defined as positive.

 

Label the angle BAP as θ.

 

Since P(x,y) := P( r cos(θ), r sin(θ) ), where ":=" means "is defined as", then:

 

sin(θ) = y/r

cos(θ) = x/r

tan(θ) := sin(θ)/cos(θ) = y/x

cot(θ) := 1/tan(θ) = x/y

sec(θ) := 1/cos(θ) = r/x

csc(θ) := 1/sin(θ) = r/y

 

Let's substitute these values into the original problem:

 

sin(θ)*cox(θ)*(tan(θ) + cot(θ)) =

(y/r)*(x/r)*(y/x + x/y) =

(xy/r^2)*( (y^2 + x^2)/(xy)) =

(xy/r^2)*( r^2/(xy)) =

1

 

The answer is One.