9.4 Sum-to-Product and Product-to-Sum Formulas

Write the following product of cosines as a sum: $2 \cos \left(\frac{7 x}{2}\right) \cos \frac{3 x}{2} .$

Use the product-to-sum formula to write the product as a sum or difference: $\cos \left(2 \theta\right) \cos \left(4 \theta\right) .$

Express the following product as a sum containing only sine or cosine and no products: $\sin \left(4 \theta\right) \cos \left(2 \theta\right) .$

Use the product-to-sum formula to write the product as a sum: $\sin \left(x + y\right) \cos \left(x - y\right) .$

Write $\cos \left(3 \theta\right) \cos \left(5 \theta\right)$ as a sum or difference.

Use the product-to-sum formula to evaluate $\cos \frac{11 \pi}{12} \cos \frac{\pi}{12} .$

Write the following difference of sines expression as a product: $\sin \left(4 \theta\right) - \sin \left(2 \theta\right) .$

Use the sum-to-product formula to write the sum as a product: $\sin \left(3 \theta\right) + \sin \left(\theta\right) .$

Evaluate $\cos \left(15°\right) - \cos \left(75°\right) .$ Check the answer with a graphing calculator.

Prove the identity:

Verify the identity $\csc^{2} \theta - 2 = \frac{\cos \left(2 \theta\right)}{\sin^{2} \theta} .$

Verify the identity $\tan \theta \cot \theta - \cos^{2} \theta = \sin^{2} \theta .$

Starting with the product to sum formula $\sin \alpha \cos \beta = \frac{1}{2} \left[\sin \left(\alpha + \beta\right) + \sin \left(\alpha - \beta\right)\right] ,$ explain how to determine the formula for $\cos \alpha \sin \beta .$

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

Rewrite the product as a sum or difference.

$16 \sin \left(16 x\right) \sin \left(11 x\right)$

Rewrite the product as a sum or difference.

$2 \sin \left(5 x\right) \cos \left(3 x\right)$

Rewrite the product as a sum or difference.

$\sin \left(- x\right) \sin \left(5 x\right)$

Rewrite the sum or difference as a product.

$\cos \left(6 t\right) + \cos \left(4 t\right)$

Rewrite the sum or difference as a product.

$\cos \left(7 x\right) + \cos \left(- 7 x\right)$

Rewrite the sum or difference as a product.

$\cos \left(3 x\right) + \cos \left(9 x\right)$

Evaluate the product for the following using a sum or difference of two functions. Evaluate exactly.

$\cos \left(45°\right) \cos \left(15°\right)$

Evaluate the product for the following using a sum or difference of two functions. Evaluate exactly.

$\sin \left(−345°\right) \sin \left(−15°\right)$

Evaluate the product for the following using a sum or difference of two functions. Evaluate exactly.

$\sin \left(−45°\right) \sin \left(−15°\right)$

Evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine.

$2 \sin \left(100°\right) \sin \left(20°\right)$

Evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine.

$\sin \left(213°\right) \cos \left(8°\right)$

Rewrite the sum as a product of two functions. Leave in terms of sine and cosine.

$\sin \left(76°\right) + \sin \left(14°\right)$

Rewrite the sum as a product of two functions. Leave in terms of sine and cosine.

$\sin \left(101°\right) - \sin \left(32°\right)$

Rewrite the sum as a product of two functions. Leave in terms of sine and cosine.

$\sin \left(−1°\right) + \sin \left(−2°\right)$

Prove the identity.

$4 \sin \left(3 x\right) \cos \left(4 x\right) = 2 \sin \left(7 x\right) - 2 \sin x$

Prove the identity.

$\sin x + \sin \left(3 x\right) = 4 \sin x \cos^{2} x$

Prove the identity.

$2 \tan x \cos \left(3 x\right) = \sec x \left(\sin \left(4 x\right) - \sin \left(2 x\right)\right)$

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.

$\cos \left(58°\right) + \cos \left(12°\right)$

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.

$\cos \left(44°\right) - \cos \left(22°\right)$

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.

$\sin \left(−14° \right) \sin \left(85°\right)$

Algebraically determine whether each of the given equation is an 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.

$\frac{\cos \left(10 \theta\right) + \cos \left(6 \theta\right)}{\cos \left(6 \theta\right) - \cos \left(10 \theta\right)} = \cot \left(2 \theta\right) \cot \left(8 \theta\right)$

Algebraically determine whether each of the given equation is an 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.

$2 \cos \left(2 x\right) \cos x + \sin \left(2 x\right) \sin x = 2 \sin x$

Simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.

$\frac{\sin \left(9 t\right) - \sin \left(3 t\right)}{\cos \left(9 t\right) + \cos \left(3 t\right)}$

Simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.

$\frac{\sin \left(3 x\right) - \sin x}{\sin x}$

Simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.

$\sin x \cos \left(15 x\right) - \cos x \sin \left(15 x\right)$

Prove the following sum-to-product formulas.

$\cos x + \cos y = 2 \cos \left(\frac{x + y}{2}\right) \cos \left(\frac{x - y}{2}\right)$

Prove the identity.

$\frac{\cos \left(3 x\right) + \cos x}{\cos \left(3 x\right) - \cos x} = - \cot \left(2 x\right) \cot x$

Prove the identity.

$\frac{\cos \left(2 y\right) - \cos \left(4 y\right)}{\sin \left(2 y\right) + \sin \left(4 y\right)} = \tan y$

Prove the identity.

$\cos x - \cos \left(3 x\right) = 4 \sin^{2} x \cos x$

Prove the identity.

$\tan \left(\frac{\pi}{4} - t\right) = \frac{1 - \tan t}{1 + \tan t}$