6 Use appropriate identities to find the exact value of 1 ta

6. Use appropriate identities to find the exact value of: (1- tan (51 degrees) tan (50 degrees)) / (tan (51 degrees) + tan (50 degrees))

7. Use appropriate identities to find the exact value of: cos (75 degrees)

8. Suppose y = sin (5x) cos x - cos (5x) sin x. What is an equivalent form of y?

9. Use appropriate identities to find the exact value of: sin (43 degrees) cos (16 degrees) + cos (43 degrees) sin (16 degrees)

10. Use a graphing calculator to determine if cot (x +1) = cot x cot 1 is an identity.

6. Use appropriate identities to find the exact value of: (1- tan (51 degrees) tan (50 degrees)) / (tan (51 degrees) + tan (50 degrees))

Use the identity : tan(x +y) = (tanx +tany)/(1 - tanx tany)

So, (1- tan (51 degrees) tan (50 degrees)) / (tan (51 degrees) + tan (50 degrees)) = 1/tan(51+50)

= 1/tan101 = cot101

7. Use appropriate identities to find the exact value of: cos (75 degrees)

cos75 = cos( 90 -15) = sin15

Use the identity : cos2x = 1- 2sin^2x

sinx = sqrt[(1 -cos2x)/2]

sin15 = sqrt[ (1 - cos30)/2]

= sqrt[ ( 2- sqrt3)/4]

= sqrt(2-sqrt3)/2

8. Suppose y = sin (5x) cos x - cos (5x) sin x. What is an equivalent form of y?

Use the trig identity :

sin(x-y) = sinxcosy - sinycosx

So,sin( 5x - x) = sin5xcosx - cos5x sinx

sin5xcosx - cos5x sinx = sin4x

9. Use appropriate identities to find the exact value of: sin (43 degrees) cos (16 degrees) + cos (43 degrees) sin (16 degrees)

use the identity : sin(x+y) = sinxcosy +cosxsiny

sin (43 degrees) cos (16 degrees) + cos (43 degrees) sin (16 degrees) = sin( 43+16)

= sin59