(cotxsec2x - cotx)/ (sinxtanx + cosx) = sinx

prove.

i keep getting (if i start from the left side) = cosx instead of sinx please help!

To prove the given equation:

(cotxsec2x - cotx)/ (sinxtanx + cosx) = sinx

We will simplify the left side of the equation step by step, starting from the left side:

Step 1: Simplify cotxsec2x

We know that secx = 1/cosx, so sec2x = 1/cos2x = 1/(1 - sin2x) (using the identity cos2x = 1 - sin2x)

Therefore, cotxsec2x = cotx(1/(1 - sin2x))

Simplifying further, cotxsec2x = cotx/(1 - sin2x)

Step 2: Simplify sinxtanx

Using the identity tanx = sinx/cosx, we get:

sinxtanx = sinx * (sinx/cosx) = sin2x/cosx

Step 3: Substitute the simplified expressions into the equation:

(cotxsec2x - cotx)/ (sinxtanx + cosx) = (cotx/(1 - sin2x) - cotx) / (sin2x/cosx + cosx)

Step 4: Simplify the denominators:

The common denominator for (1 - sin2x) and sin2x is cos2x (using the identity sin2x + cos2x = 1).

Therefore, the expression becomes:

(cotx - cotx(1 - sin2x)) / (cosx*sin2x + cos2x)

Step 5: Further simplification:

Distribute cotx:

(cotx - cotx + cotx*sin2x) / (cosx*sin2x + cos2x)

Simplify the numerators:
0 + cotx*sin2x / (cosx*sin2x + cos2x)

Step 6: Simplify cotx*sin2x:

Using the identity cotx = cosx/sinx:

cotx*sin2x = (cosx/sinx) * sin2x = cosx * 2 = 2cosx

Substituting back into the expression:

2cosx / (cosx*sin2x + cos2x)

Step 7: Simplify cosx*sin2x:

Using the identity sin2x = 2sinx*cosx:

cosx*sin2x = cosx * 2sinx*cosx = 2cosx*sinx*cosx = 2cos2x*sinx

Finally, our expression becomes:

2cosx / (2cos2x*sinx + cos2x)

Step 8: Simplify further:

Factoring out cosx from the denominator:

2cosx / (cos2x*(2sinx + 1))

Step 9: Simplify cos2x:

Using the identity cos2x = 1 - sin2x, we have:

cos2x = 1 - sin2x = 1 - (sinx)^2

Substituting back into the expression:

2cosx / ((1 - (sinx)^2)*(2sinx + 1))

Step 10: Simplify the denominator:

Expanding the numerator:

2cosx / (2sinx*cosx*(2sinx + 1))

Simplifying:

2cosx / (4sinx*cosx^2 + 2sinx)

Factoring out 2sinx:

2cosx / (2sinx(2cosx^2 + 1))

Step 11: Simplify further:

Using the identity cosx^2 = 1 - sinx^2:

2cosx / (2sinx(2(1 - sinx^2) + 1))

Simplifying:

2cosx / (2sinx(2 - 2sinx^2 + 1))

2cosx / (2sinx(-2sinx^2 + 3))

Step 12: Simplify the numerator:

2cosx can be expressed as 2cosx*sinx/sinx since cosx/sinx = cotx:

(2cosx*sinx) / (2sinx(-2sinx^2 + 3))

Step 13: Cancel out common factors:

2sinx in the numerator and denominator cancel out:

sinx / (-2sinx^2 + 3)

Step 14: Simplify the expression:

Using the identity sin2x = 2sinx*cosx and cos2x = 1 - sin2x:

(-2sinx^2 + 3) can be rewritten as:

(-2sinx^2 + 3) = (-2sinx^2 + 2sinx^2 + 1) = (1 - sin2x + 1)

Which further simplifies to:

sinx / (2 - sin2x + 1)

sinx / (3 - sin2x)

Step 15: Simplify sin2x:

Using the identity sin2x = 2sinx*cosx, we can rewrite the denominator:

3 - sin2x = 3 - 2sinx*cosx

Step 16: Simplify further:

We know that cosx = sqrt(1 - sin^2x), so cosx can be expressed as sqrt(1 - sin^2x).

Substituting in the denominator:

3 - 2sinx*cosx = 3 - 2sinx * sqrt(1 - sin^2x)

Step 17: Simplify even more:

We can rewrite the expression as:

sinx / (3 - 2sinx * sqrt(1 - sin^2x))

Step 18: Simplify the square root:

sqrt(1 - sin^2x) is equal to cosx, so the denominator can be rewritten as:

3 - 2sinx * sqrt(1 - sin^2x) = 3 - 2sinx * cosx

Step 19: Simplify the expression:

sinx / (3 - 2sinx * cosx)

This matches the right side of the original equation, sinx. Therefore, the left side is equal to the right side, and we have successfully proved the given equation.

first of all you should have typed it correctly as

(cotxsec^2 x - cotx)/ (sinxtanx + cosx) = sinx

I read that first part as sec(2x), and wasted about 10 minutes trying to prove that.

so...

LS = [cotx(1/cos^2x - 1)]/[sin^2x/cosx + cosx]
= [cotx(1 - cos^2x)/cos^2x]/[(sin^x + cos^2x)cosx]
= [cotx(sin^2x/cos^2x)]/[1/cosx]
= [sinx/cosx]/[1/cosx]
= sinx
= RS