[Note: I'm still having issues with identities with regard to trig]

Verify the Trig. identities:

(a). cot(x+y)=(cotxcoty-1)/(cotx+coty)

(b). sin0(cot0+tan0)=sec0

[Note: 0=theta symbol]

recall that

tan(x+y) = (tanx + tany)/(1-tanx*tany)

so,

cot(x+y) = (1 - tanx tany)/(tanx + tany)

Now divide top and bottom by tanx*tany

sinx(cotx+tanx)
= sinx(cosx/sinx + sinx/cosx)
= cosx + sin^2x/cosx
= (cos^2x + sin^2x)/cosx
= secx

To verify the given trigonometric identities, we need to manipulate one side of the equation until it simplifies to the other side.

(a) cot(x+y) = (cot(x)cot(y) - 1)/(cot(x) + cot(y))

To start, let's convert the cot(x+y) to its equivalent expression using cosine and sine:

cot(x+y) = cot(x)cot(y) - 1 / sin(x+y) / cos(x+y)

Since sin(x+y) = sin(x)cos(y) + cos(x)sin(y) and cos(x+y) = cos(x)cos(y) - sin(x)sin(y), we can substitute these values into the equation:

cot(x+y) = cot(x)cot(y) - 1 / (sin(x)cos(y) + cos(x)sin(y)) / (cos(x)cos(y) - sin(x)sin(y))

Next, simplify the equation by combining the fractions:

cot(x+y) = (cot(x)cot(y) - 1) / [(sin(x)cos(y) + cos(x)sin(y)) / (cos(x)cos(y) - sin(x)sin(y))]

To further simplify the equation, you can multiply the numerator and denominator by (cos(x)cos(y) - sin(x)sin(y)) to get rid of the fraction in the denominator:

cot(x+y) = (cot(x)cot(y)-1) * (cos(x)cos(y) - sin(x)sin(y))/(sin(x)cos(y) + cos(x)sin(y))

Now, expand the expression:

cot(x+y) = (cot(x)cot(y)cos(x)cos(y) - cot(x)cot(y)sin(x)sin(y) - cos(x)cos(y) + sin(x)sin(y))/(sin(x)cos(y) + cos(x)sin(y))

Combine the like terms:

cot(x+y) = (cot(x)cot(y)cos(x)cos(y) - cos(x)cos(y) - cot(x)cot(y)sin(x)sin(y) + sin(x)sin(y))/(sin(x)cos(y) + cos(x)sin(y))

At this point, we can observe that the numerator can be factored:

cot(x+y) = [cos(x)cos(y)(cot(x)cot(y) - 1) - sin(x)sin(y)(cot(x)cot(y) - 1)] / (sin(x)cos(y) + cos(x)sin(y))

Notice that (cot(x)cot(y) - 1) appears in both terms of the numerator:

cot(x+y) = (cos(x)cos(y) - sin(x)sin(y))(cot(x)cot(y) - 1) / (sin(x)cos(y) + cos(x)sin(y))

Now, we can simplify further by canceling out the common factor (cot(x)cot(y) - 1) in the numerator and denominator:

cot(x+y) = (cos(x)cos(y) - sin(x)sin(y)) / 1

Finally, we obtain:

cot(x+y) = cos(x)cos(y) - sin(x)sin(y)

Therefore, the identity cot(x+y) = cos(x)cos(y) - sin(x)sin(y) holds.

(b) sin(theta)(cot(theta) + tan(theta)) = sec(theta)

To verify this trigonometric identity, we'll start with the left-hand side:

sin(theta)(cot(theta) + tan(theta))

Using the definitions of cotangent and tangent, we can replace them in terms of sine and cosine:

= sin(theta)(cos(theta)/sin(theta) + sin(theta)/cos(theta))

Simplify the expression by finding the common denominator:

= sin(theta)(cos^2(theta) + sin^2(theta))/(sin(theta)cos(theta))

Since cos^2(theta) + sin^2(theta) = 1, the numerator simplifies to:

= sin(theta)/sin(theta)cos(theta)

Next, cancel out the common factor of sin(theta) in the numerator and denominator:

= 1/cos(theta)

Finally, recall that the reciprocal of cosine is secant, so:

= sec(theta)

Therefore, the identity sin(theta)(cot(theta) + tan(theta)) = sec(theta) holds.