For the following exercises, rewrite the sum as a product of two functions or the

Starting with the product to sum formula \(\sin \alpha \cos \beta=\frac{1}{2}[\sin (\alpha+\beta)+\sin (\alpha-\beta)]\), explain how to determine the formula for cos ? sin ?. Text Transcription: sin ? cos ? = ½ [sin(? + ?) + sin(? ? ?)]

For the following exercises, prove the identity. \(\frac{\cos (a+b)}{\cos (a-b)}=\frac{1-\tan a \tan b}{1+\tan a \tan b}\) Text Transcription: cos (a + b) / cos (a - b) = 1 - tan a tan b / 1 + tan a tan b

For the following exercises, prove the identity. \(\frac{6 \cos (8 x) \sin (2 x)}{\sin (-6 x)}=-3 \sin (10 x) \csc (6 x)+3\) Text Transcription: 6cos(8x) sin(2x) / sin (-6x) = -3 sin(10x) csc(6x) + 3

For the following exercises, prove the identity. \(sin x + sin(3x) = 4sin x cos^{2} x\) Text Transcription: sin x + sin(3x) = 4sin x cos^2 x

For the following exercises, prove the identity. \(2(cos^{3} x ? cos x sin^{2} x)= cos(3x) + cos x\) Text Transcription: 2(cos^3 x ? cos x sin^2 x)= cos(3x) + cos x

For the following exercises, algebraically determine whether each of the given expressions is a true identity. If it is not an identity, replace the right-hand side with an expression equivalent to the left side. Verify the results by graphing both expressions on a calculator. \(\frac{\cos (10 \theta)+\cos (6 \theta)}{\cos (6 \theta)-\cos (10 \theta)}=\cot (2 \theta) \cot (8 \theta)\) Text Transcription: cos(10?) + cos (6?) / cos (6?) - cos (10?) = cot(2?)cot(8?)

For the following exercises, algebraically determine whether each of the given expressions is a true identity. If it is not an identity, replace the right-hand side with an expression equivalent to the left side. Verify the results by graphing both expressions on a calculator. \(\frac{\sin (3 x)-\sin (5 x)}{\cos (3 x)+\cos (5 x)}=\tan x\) Text Transcription: sin(3x) - sin(5x) / cos(3x) + cos(5x) = tan x

For the following exercises, algebraically determine whether each of the given expressions is a true identity. If it is not an identity, replace the right-hand side with an expression equivalent to the left side. Verify the results by graphing both expressions on a calculator. \(\frac{\sin (2 x)+\sin (4 x)}{\sin (2 x)-\sin (4 x)}=-\tan (3 x) \cot x\) Text Transcription: sin(2x) + sin (4x) / sin(2x) - sin(4x) = tan(3x) cot x

For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical. \(\frac{\sin (9 t)-\sin (3 t)}{\cos (9 t)+\cos (3 t)}\) Text Transcription: sin(9t) - sin(3t) / cos(9t) + cos(3t)

For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical. \(\frac{\sin (3 x)-\sin x}{\sin x}\) Text Transcription: sin (3x) - sin x / sin x

For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical. \(\frac{\cos (5 x)+\cos (3 x)}{\sin (5 x)+\sin (3 x)}\) Text Transcription: cos(5x) + cos (3x) / sin(5x) + sin(3x)

For the following exercises, prove the following sum-to-product formulas. \(\sin x-\sin y=2 \sin \left(\frac{x-y}{2}\right) \cos \left(\frac{x+y}{2}\right)\) Text Transcription: sin x ? sin y = 2 sin (x - y / 2) cos (x + y / 2)

For the following exercises, prove the following sum-to-product formulas. \(\cos x+\cos y=2 \cos \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right)\) Text Transcription: cos x + cos y = 2cos (x + y / 2) cos (x - y / 2)

For the following exercises, prove the identity. \(\frac{\sin (6 x)+\sin (4 x)}{\sin (6 x)-\sin (4 x)}=\tan (5 x) \cot x\) Text Transcription: sin (6x) + sin (4x) / sin(6x) - sin(4x) = tan (5x) cot x

For the following exercises, prove the identity. \(\frac{\cos (3 x)+\cos x}{\cos (3 x)-\cos x}=-\cot (2 x) \cot x\) Text Transcription: cos (3x) + cos x / cos(3x) - cos x = cot (2x) cot x

For the following exercises, prove the identity. \(\frac{\cos (6 y)+\cos (8 y)}{\sin (6 y)-\sin (4 y)}=\cot y \cos (7 y) \sec (5 y)\) Text Transcription: cos (6y) + cos (8y) / sin (6y) - sin (4y) = cot y cos(7y) sec(5y)

For the following exercises, prove the identity. \(\frac{\cos (2 y)-\cos (4 y)}{\sin (2 y)+\sin (4 y)}=\tan y\) Text Transcription: cos(2y) - cos(4y) / sin(2y) + sin(4y) = tan y

For the following exercises, prove the identity. \(\frac{\sin (10 x)-\sin (2 x)}{\cos (10 x)+\cos (2 x)}=\tan (4 x)\) Text Transcription: sin (10x) - sin (2x) / cos (10x) + cos (2x) = tan (4x)

For the following exercises, prove the identity. \(cos x ? cos(3x) = 4 sin^{2} x cos x\) Text Transcription: cos x ? cos(3x) = 4 sin^2 x cos x

For the following exercises, prove the identity. \((cos(2x) ? cos(4x))^{2} + (sin(4x) + sin(2x))^{2} = 4 sin^{2}(3x)\) Text Transcription: (cos(2x) ? cos(4x))^2 + (sin(4x) + sin(2x))^2 = 4 sin^2(3x)

For the following exercises, prove the identity. \(\tan \left(\frac{\pi}{4}-t\right)=\frac{1-\tan t}{1+\tan t}\) Text Transcription: tan (? / 4 - t ) = 1 - tan t / 1 + tan t

Provide two different methods of calculating cos(195°)cos(105°), one of which uses the product to sum. Which method is easier?

Describe a situation where we would convert an equation from a sum to a product and give an example.

Describe a situation where we would convert an equation from a product to a sum, and give an example.

For the following exercises, rewrite the product as a sum or difference. 16sin(16x)sin(11x)

For the following exercises, rewrite the product as a sum or difference. 20cos(36t)cos(6t)

For the following exercises, rewrite the product as a sum or difference. 2sin(5x)cos(3x)

For the following exercises, rewrite the product as a sum or difference. 10cos(5x)sin(10x)

For the following exercises, rewrite the product as a sum or difference. sin(?x)sin(5x)

For the following exercises, rewrite the product as a sum or difference. sin(3x)cos(5x)

For the following exercises, rewrite the sum or difference as a product. cos(6t) + cos(4t)

For the following exercises, rewrite the sum or difference as a product. sin(3x) + sin(7x)

For the following exercises, rewrite the sum or difference as a product. cos(7x) + cos(?7x)

For the following exercises, rewrite the sum or difference as a product. sin(3x) ? sin(?3x)

For the following exercises, rewrite the sum or difference as a product. cos(3x) + cos(9x)

For the following exercises, rewrite the sum or difference as a product. sin h ? sin(3h)

For the following exercises, evaluate the product using a sum or difference of two functions. Evaluate exactly. cos(45°)cos(15°)

For the following exercises, evaluate the product using a sum or difference of two functions. Evaluate exactly. cos(45°)sin(15°)

For the following exercises, evaluate the product using a sum or difference of two functions. Evaluate exactly. sin(?345°)sin(?15°)

For the following exercises, evaluate the product using a sum or difference of two functions. Evaluate exactly. sin(195°)cos(15°)

For the following exercises, evaluate the product using a sum or difference of two functions. Evaluate exactly. sin(?45°)sin(?15°)

For the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine. cos(23°)sin(17°)

For the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine. 2sin(100°)sin(20°)

For the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine. 2sin(?100°)sin(?20°)

For the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine. sin(213°)cos(8°)

For the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine. 2cos(56°)cos(47°)

For the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine. sin(76°) + sin(14°)

For the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine. cos(58°) ? cos(12°)

For the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine. sin(101°) ? sin(32°)

For the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine. cos(100°) + cos(200°)

For the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine. sin(?1°) + sin(?2°)

For the following exercises, prove the identity. 4sin(3x)cos(4x) = 2 sin(7x) ? 2 sinx

For the following exercises, prove the identity. 2 tan x cos(3x) = sec x(sin(4x) ? sin(2x))

For the following exercises, prove the identity. cos(a + b) + cos(a ? b) = 2cos a cos b

For the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places. cos(58°) + cos(12°)

For the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places. sin(2°) ? sin(3°)

For the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places. cos(44°) ? cos(22°)

For the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places. cos(176°)sin(9°)

For the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places. sin(?14°)sin(85°)

For the following exercises, algebraically determine whether each of the given expressions is a true identity. If it is not an identity, replace the right-hand side with an expression equivalent to the left side. Verify the results by graphing both expressions on a calculator. 2sin(2x)sin(3x) = cos x ? cos(5x)

"For the following exercises, algebraically determine whether each of the given expressions is a true identity. If it is not an identity, replace the right-hand side with an expression equivalent to the left side. Verify the results by graphing both expressions on a calculator. 2cos(2x)cos x + sin(2x)sin x = 2 sin x"

For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical. 2sin(8x)cos(6x) ? sin(2x)

"For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical. sin x cos(15x) ? cos x sin(15x)"