Proving Trigonometric Identities

In summary, we are given four trigonometric equations and are asked to manipulate them using various trigonometric identities. The first equation involves the product of sines and the difference of cosines. The second equation has two tangent functions and a secant function. The third equation has a difference of cosines and a product of sines, while the fourth equation has a cotangent, tangent, and secant function. To solve these equations, we will need to use the tangent sum and difference formulas.
  • #1
mathnewb
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Homework Statement


1) sin(x+y)sin(x-y)=cos^2y-cos^2x

2) tan(∏⁄4+x)+tan(∏/4-x)=2sec2x

3) cosx-cosy=-2sin(x+y/2)sin(x+y/2)

4) 2cotx-2tanx=4-2sec^2x/tanx

Homework Equations


all trig identities

The Attempt at a Solution


1) i understand that i should show what i have attempted but there is way too many lines of manipulating the equations and i don't have the time to write it down. i can say that for all of these identities i have changed them to cos and sin as these can be the easiest to solve and then used trig identities to make both sides look the same. i have nt had any luck with these and i was hoping you guys could help me

thx alot!
 
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  • #2
3) cosx-cosy=-2sin(x+y/2)sin(x+y/2)
Is it (x+y)/2 or x+(y/2)?
4) 2cotx-2tanx=4-2sec^2x/tanx
Again... is it (4-2sec^2x)/tanx or 4-(2sec^2x/tanx)?
1) i understand that i should show what i have attempted but there is way too many lines of manipulating the equations and i don't have the time to write it down. i can say that for all of these identities i have changed them to cos and sin as these can be the easiest to solve and then used trig identities to make both sides look the same. i have nt had any luck with these and i was hoping you guys could help me
Sorry, you'll have to make time to write what you have so far if you want us to help. I'll give you one suggestion -- in #2, don't convert to cosines and sines as your first step. Too messy. Instead, use the tangent sum & difference formulas.01
 
Last edited:
  • #3


Hi there,

I understand that proving trigonometric identities can be challenging and time-consuming. It requires a good understanding of the fundamental trigonometric functions and their properties. I would suggest breaking down each identity into smaller steps and using known trigonometric identities to manipulate the expressions. For example, for the first identity, we can use the double angle formula for cosine to expand cos²y and cos²x. Then, we can use the sum and difference formula for sine to expand sin(x+y) and sin(x-y). After that, we can rearrange the terms to show that both sides are equal.

For the second identity, we can use the sum and difference formula for tangent to expand both terms on the left side. Then, we can use the reciprocal identities for tangent and cosine to simplify the expression further. Finally, we can use the double angle formula for cosine to show that both sides are equal.

For the third identity, we can use the half-angle formula for cosine to expand cosx and cosy. Then, we can use the sum formula for sine to expand sin(x+y/2) twice. After that, we can rearrange the terms to show that both sides are equal.

For the fourth identity, we can use the reciprocal identities for cotangent and tangent to simplify the expression. Then, we can use the double angle formula for sine to expand sin2x. Finally, we can use the Pythagorean identity to show that both sides are equal.

I hope this helps and gives you a starting point for solving these identities. Remember to use known trigonometric identities and manipulate the expressions step by step. Good luck!
 

What are trigonometric identities?

Trigonometric identities are mathematical equations that involve trigonometric functions such as sine, cosine, and tangent. These identities are used to simplify and solve complex trigonometric expressions.

Why is it important to prove trigonometric identities?

Proving trigonometric identities is important because it helps to verify the validity of equations and allows for the simplification of complex expressions. It also helps in understanding the relationships between different trigonometric functions.

What are the steps to prove a trigonometric identity?

The steps to prove a trigonometric identity involve manipulating the given equation using algebraic techniques, such as factoring and expanding, and applying trigonometric identities and properties, such as the Pythagorean identity and double angle formulas. The goal is to transform one side of the equation to match the other side.

What are some common strategies for proving trigonometric identities?

Some common strategies for proving trigonometric identities include using fundamental identities, converting all trigonometric functions to sine and cosine, and working with one side of the equation at a time. It is also helpful to use symmetry and substitution to simplify expressions.

Are there any tips for successfully proving trigonometric identities?

Some tips for successfully proving trigonometric identities include being familiar with common trigonometric identities, practicing solving similar problems, and carefully checking for errors. It is also important to have a strong foundation in algebra and trigonometry.

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