Cos^2 - Sin^2 = Cos(2a): Shaum's Solution

  • Thread starter Ai52487963
  • Start date
  • Tags
    identities
In summary, the difference of cos^2 and sin^2 is a well known identity, cos^2 - sin^2 = cos(2a). This is a part of a more general identity, cos(x+y) = cos(x)cos(y) - sin(x)sin(y). This can be used to find exact angles using complex numbers and prove other double, triple, and quadruple angle identities. Another famous identity is cos^2 + sin^2 = 1, also known as the Pythagorean identity, which can be used to derive two other forms using secant and cosecant.
  • #1
Ai52487963
115
0
Anyone know if the difference of cos^2 and sin^2 is some obscure identity that no one's heard of?

Edit: nevermind. Shaum's tells me that cos^2 - sin^2 = cos(2a). GO SHAUMS!
 
Last edited:
Mathematics news on Phys.org
  • #2
So the answer to your question is "NO"! It is, in fact, a well known identity!

A more general identity is cos(x+ y)= cos(x)cos(y)- sin(x)sin(y). Letting x= y= a in that, cos(2a)= cos2(a)- sin2(a).
 
  • #3
If you know complex numbers [including de Moivre's theorem (great chap, wasn't he) and Binomial Theorem], you can find the exact angle of any sin/cos/tan in surd form, and you can prove any double angle/triple angle/quadruple angle etc angles.

Back on topic though, yes those identities are very famous, and cos^2+sin^2 = 1 is also a darned famous one, think Pythagoras. From that you can get 2 more forms, one with sec and the other with csc
 

FAQ: Cos^2 - Sin^2 = Cos(2a): Shaum's Solution

What is the formula for Cos^2 - Sin^2 = Cos(2a)?

The formula for Cos^2 - Sin^2 = Cos(2a) is a trigonometric identity that states the difference between the squares of the cosine and sine functions is equal to the cosine of twice the angle (2a).

How is "Cos^2 - Sin^2 = Cos(2a)" derived?

The formula is derived using the double-angle formula for cosine. In this case, we substitute 2a for the angle, giving us Cos(2a) = Cos^2 a - Sin^2 a. Rearranging this equation gives us Cos^2 a - Sin^2 a = Cos(2a).

What is the significance of "Cos^2 - Sin^2 = Cos(2a)" in mathematics?

Trigonometric identities, such as Cos^2 - Sin^2 = Cos(2a), are important in mathematics because they allow us to simplify complicated expressions and solve equations involving trigonometric functions. They also have applications in various fields such as physics, engineering, and astronomy.

How can "Cos^2 - Sin^2 = Cos(2a)" be used in real-world situations?

This formula can be used to solve problems involving triangles and circles, such as finding the missing side or angle in a right triangle. It can also be applied in physics to calculate the displacement, velocity, or acceleration of objects moving in circular paths.

Are there any other similar trigonometric identities?

Yes, there are many other trigonometric identities that involve the cosine and sine functions, including double angle, half angle, sum and difference, and product-to-sum identities. These identities can be derived from basic trigonometric relationships and have various uses in mathematics and other fields.

Similar threads

B Trigonometric Identity involving sin()+cos()
Replies
11
Views
1K
I Determine ## \beta ## as a function of ##\theta## linkage
Replies
2
Views
1K
MHB How Can You Prove the Trigonometric Identity Cos^6A+Sin^6A=1-3Sin^2ACos^2A?
Replies
6
Views
2K
MHB Solving trig equation cos(x)=sin(x) + 1/√3
Replies
3
Views
3K
B To calculate the polar unit vectors using rotated coordinates
Replies
1
Views
172
MHB Solving Trig Identity: Sin[1/2 sin^−1 (x)] = 1/2 X (sqr(1+x) –sqr(1-x))
Replies
2
Views
1K
B Why do we care about trig identities?
Replies
17
Views
5K
MHB Trigonometric Identity: Tan^2-Sin^2 = Sin^2 Cos^2
Replies
2
Views
2K
MHB Proving Identity 46: $\cos^{6}(A)+\sin^{6}(A)=1-3\sin^{2}(A)\cos^{2}(A)$
Replies
2
Views
2K
MHB Evaluate cos 2(theta) and sin 2(theta)
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top