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# Trigonometry

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1) Verify the following identities :

a) sin(x+y)cos(x-y) + cos(x+y)sin ( x-y) = sin 2x

b) cos2x = [cot^2 (x-1 )] / [ cot^2 (x+1) ]

2) Derive the identity for sin 3x in terms of sin x

3) Using the double-angle formula, find sin 120° .

4)Simplify the following expressions so that they involve a function of only one angle:

a) (sin 80° - sin 10° ) / (sin 80° + sin 10°)
b) (sin 130° + sin 20°) / (cos 130° + cos 20 °)

A) Use logarithms and the law of tangents to solve the triangle ABC,
given that a=21.46 ft, b=46.28 ft, and C=32° 28' 30"

B) Solve the triangle for which the given parts are :

a=27, b=21, and c=24.

https://brainmass.com/math/trigonometry/plane-trigonometry-verifying-identities-187514

#### Solution Preview

The solution file is attached.

1) Verify the following identities :

a) sin(x+y)cos(x-y) + cos(x+y)sin ( x-y) = sin 2x

LHS = sin(x + y) cos(x - y) + cos(x + y) sin(x - y)
= sin A cos B + cos A sin B [where x + y = A and x - y = B]
= sin (A + B)
= sin[(x + y) + (x - y)]
= sin 2x
= RHS

b) cos2x = [cot^2 (x-1 )] / [ cot^2 (x+1) ]

There appears to be some error in this question. This identity does not hold for x = 1 (for example).
When x = 1, LHS = cos (2 * 1) = cos 2 (finite), whereas,
RHS = [cot^2 (1 - 1)]/[cot^2 (1 + 1)] = cot^2 (0) / cot^2 (2), which is undefined.