Question

Sin^2a cos^2B-cos^2a sin^2B+cos^2a

Answer 1

Sin²αCos²β - Cos²αSin²β + Cos²α=
=Sin²αCos²β +Cos²α(1 - Sin²β) = Sin²αCos²β+ Cos²αCos²β=
=Cos²β(Sin²α + Cos²α) = Cos²β·1= Cos²β

Answer 2

To simplify the expression sin^2(a) * cos^2(B) - cos^2(a) * sin^2(B) + cos^2(a), you can use the trigonometric identities for the product of sines and cosines. One of these identities is:

sin(2θ) = 2sin(θ)cos(θ)

You can use this identity to simplify your expression:

sin^2(a) * cos^2(B) - cos^2(a) * sin^2(B) + cos^2(a)

= sin^2(a) * cos^2(B) - cos^2(a) * sin^2(B) + cos^2(a) * (1 - sin^2(a))

Now, expand and rearrange the terms:

= sin^2(a) * cos^2(B) - cos^2(a) * sin^2(B) + cos^2(a) - cos^2(a) * sin^2(a)

Now, apply the identity sin(2θ) = 2sin(θ)cos(θ) to the first two terms:

= 2sin(a)cos(a) * 2sin(B)cos(B) + cos^2(a) - cos^2(a) * sin^2(a)

Now, use the double angle identity for sine, which is sin(2θ) = 2sin(θ)cos(θ):

= 4sin(a)cos(a)sin(B)cos(B) + cos^2(a) - cos^2(a) * sin^2(a)

Now, use the identity sin^2(θ) + cos^2(θ) = 1:

= 4sin(a)cos(a)sin(B)cos(B) + (1 - sin^2(a)) - cos^2(a) * sin^2(a)

= 4sin(a)cos(a)sin(B)cos(B) + 1 - sin^2(a) - cos^2(a) * sin^2(a)

Since sin^2(a) + cos^2(a) = 1, you can substitute 1 for sin^2(a) + cos^2(a):

= 4sin(a)cos(a)sin(B)cos(B) + 1 - 1 * sin^2(a)

= 4sin(a)cos(a)sin(B)cos(B) + 1 - sin^2(a)

So, the simplified expression is:

4sin(a)cos(a)sin(B)cos(B) + 1 - sin^2(a)