1.

18 + 72 = 90, and 36 + 54 = 90.

cos(90 – x) = sin(x).

Therefore:

sin^2(18) + sin^2(72) + sin^2(36) + sin^2(54)

= [sin^2(18) + cos^2(18)] + [sin^2(36) + cos^2(36)]

= 1 + 1

= 2.

2.

I don’t understand this question.

3.

sin6 x + cos6 x -2sin4 x -cos4 x + sin2 x

[sin^6(x) – 2sin^4(x) + sin^2(x)] + [cos^6(x) – cos^4(x)]

= sin^2(x)[sin^4(x) – 2sin^2(x) + 1] + cos^4(x)[cos^2(x) – 1]

= sin^2(x)[sin^2(x) – 1]^2 – cos^4(x)sin^2(x)

= sin^2(x)cos^4(x) – cos^4(x)sin^2(x)

= 0.

4.

(cot2 x) (cos2 x) + cot2 x – cos2 x

= cot^2(x)cos^2(x) + cot^2(x) – cot^2(x)sin^2(x)

= cot^2(x)[ cos^2(X) + {1 – sin^2(x)} ]

= 2cot^2(x) cos^2(x).

5.

Same principle as question 1.

sin2 (x+ 10) + cos2 (20 – x ) + sin2 (80 – x) + cos2 (70 + x)

= [sin^2(x + 10) + sin^2(80 – x)] + [cos^2(20 – x ) + cos^2(70 + x]

= [sin^2(x + 10) + cos^2(x + 10)] + [sin^2(70 + x) + cos^2(70 + x)]

= 1 + 1

= 2.

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