Question

Prove the identity.

cos2x - sin’2x = 2 cos2x - 1
Note that each Statement must be based on a Rule chosen from the Rule menu.
cos2x - sin’2x
A. Algebra.
B. Reciprocal.
C. Quotient.
D. Pythagorean.
E. Odd/Even.

Answer 1

D. Pythagorean

Explanation:

Given the identity

cos²x - sin²x = 2 cos²x - 1.

To show that the identity is true, we need to show that the left hand side is equal to right hand side or vice versa.

Starting from the left hand side

cos²x - sin²x ... 1

According to Pythagoras theorem, we know that x²+y² = r² in a right angled triangle. Coverting this to polar form, we have:

x = rcostheta

y = rsintheta

Substituting into the Pythagoras firnuka we have

(rcostheta)²+(rsintheta)² = r²

r²cos²theta+r²sin²theta = r²

r²(cos²theta+sin²theta) = r²

(cos²theta+sin²theta) = 1

sin²theta = 1 - cos²theta

sin²x = 1-cos²x ... 2

Substituting equation 2 into 1 we have;

= cos²x-(1-cos²x)

= cos²x-1+cos²x

= 2cos²x-1 (RHS)

This shows that cos²x -sin²x = 2cos²x-1 with the aid of PYTHAGORAS THEOREM