Trigonometry identities and equations

In summary, to simplify 2sec^2x-2sec^2xsin^2x-sin^2x-cos^2x, we can use the equation sec(x)=1/cos(x) to rewrite it as 2(1/cos^2x)-2(1/cos^2x)sin^2x-sin^2x-cos^2x. Then, we can combine like terms to get 2-2sin^2x-2/cos^2x-sin^2x-cos^2x. Finally, using the Pythagorean identity sin^2x+cos^2x=1 and the equation sec(x)=1/cos(x), we
  • #1
wei1006
6
0
1) Question statement:
Simplify 2sec^2x-2sec^2xsin^2x-sin^2x-cos^2x

2)Relevant equations:
tan A=sinA/cos A
1+tan^2A=sec^A
cot A=1/tanA
cot A=cos A/sinA
sin^2A+cos^2A=1
secA=1/cos A
cosecA=1/sinA
1+cosec^2A= cot^2A
sin2A=2sinAcosA
cos2A=1-2sin^2A=cos^2A-sin^2A=2cos^A-1
tan2A=(2tanA)/1-tan^2A

3) Attempt:
2sec^2x-2sec^2xsin^2x-sin^2x-cos^2x
= 2sec^2x(1-sin^2x)-sin^2x-cos^2x
= 2sec^2x(cos^2x)-sin^2x-cos^2x

I am stuck after this... Will be helpful if some clues are provided. Thank you!
 
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  • #2
wei1006 said:
1) Question statement:
Simplify 2sec^2x-2sec^2xsin^2x-sin^2x-cos^2x

2)Relevant equations:
tan A=sinA/cos A
1+tan^2A=sec^A
cot A=1/tanA
cot A=cos A/sinA
sin^2A+cos^2A=1
secA=1/cos A
cosecA=1/sinA
1+cosec^2A= cot^2A
sin2A=2sinAcosA
cos2A=1-2sin^2A=cos^2A-sin^2A=2cos^A-1
tan2A=(2tanA)/1-tan^2A

3) Attempt:
2sec^2x-2sec^2xsin^2x-sin^2x-cos^2x
= 2sec^2x(1-sin^2x)-sin^2x-cos^2x
= 2sec^2x(cos^2x)-sin^2x-cos^2x

I am stuck after this... Will be helpful if some clues are provided. Thank you!

Use that sec(x)=1/cos(x).
 
  • #3
Also, at the end you have: (... -sin^2x - cos^2x) . What could you do with that?
 

Related to Trigonometry identities and equations

1. What are the basic trigonometric identities?

There are six basic trigonometric identities: sine, cosine, tangent, cotangent, secant, and cosecant. These identities are used to relate the sides and angles of a right triangle.

2. How do I solve trigonometric equations?

To solve a trigonometric equation, you need to use the trigonometric identities and properties to simplify the equation and isolate the variable. You may also need to use algebraic techniques, such as factoring or substitution, to solve the equation.

3. What is the Pythagorean identity?

The Pythagorean identity is one of the most well-known trigonometric identities. It states that for any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In other words, a2 + b2 = c2, where c is the length of the hypotenuse and a and b are the lengths of the other two sides.

4. How do I prove trigonometric identities?

To prove a trigonometric identity, you need to manipulate one side of the equation using the properties and identities of trigonometric functions until it becomes equal to the other side. This can involve using basic trigonometric identities, simplifying or expanding expressions, or using other algebraic techniques.

5. Why are trigonometric identities important?

Trigonometric identities are important because they allow us to solve equations involving trigonometric functions, simplify complicated expressions, and prove other mathematical concepts. They are also used in various fields, such as physics, engineering, and navigation, to solve real-world problems that involve angles and triangles.

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