Question

Sin^2 (pi/4+a)- sin^2 (pi/4-a)

Answer 1

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Answer 2

We can start by using the trigonometric identity:

sin^2 (x) + cos^2 (x) = 1

This can be rearranged to get:

sin^2 (x) = 1 - cos^2 (x)

Now, let's apply this identity to the expression we want to simplify:

sin^2 (π/4 + a) - sin^2 (π/4 - a)

= [1 - cos^2(π/4 + a)] - [1 - cos^2(π/4 - a)]

= cos^2(π/4 - a) - cos^2(π/4 + a)

Next, we can use another trigonometric identity:

cos (a + b) = cos(a)cos(b) - sin(a)sin(b)

Using this identity, we can rewrite the expression as:

cos^2(π/4 - a) - cos^2(π/4 + a)

= [cos(π/4 - a) + cos(π/4 + a)][cos(π/4 - a) - cos(π/4 + a)]

= [cos(π/4)cos(a) - sin(π/4)sin(a) + cos(π/4)cos(a) + sin(π/4)sin(a)][cos(π/4 - a) - cos(π/4 + a)]

= [√2cos(a) + 0][cos(π/4 - a) - cos(π/4 + a)]

= √2cos(a)[cos(π/4 - a) - cos(π/4 + a)]

Now, we can use another trigonometric identity:

cos (a - b) = cos(a)cos(b) + sin(a)sin(b)

Using this identity, we can simplify the expression further:

√2cos(a)[cos(π/4 - a) - cos(π/4 + a)]

= √2cos(a)[cos(π/4)cos(a) + sin(π/4)sin(a) - cos(π/4)cos(a) + sin(π/4)sin(a)]

= √2cos(a)[2sin(π/4)sin(a)]

= √2cos(a)[√2/2sin(a)]

= 2cos(a)

Therefore, the simplified expression is 2cos(a).