Proving Trigonometric Identities

[mathnewb]

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!

 

[yeongil]

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

 

[Architeuthis Dux]

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!