1) Let us show that
a) sin66° – cos24° = 0
b) cos²57° + cos²33° = 1
c) cos²75° – sin²15° = 0
d) cosec²48° – ta

2 thoughts on “1) Let us show that<br />a) sin66° – cos24° = 0<br />b) cos²57° + cos²33° = 1<br />c) cos²75° – sin²15° = 0<br />d) cosec²48° – ta”

1. Answer:

   Question :-

   a) sin66° – cos24° = 0

   b) cos²57 + cos²33° = 1

   c) cos²75° – sin²15° = 0

   d) cosec²48° – tan²42° = 1

   e) sec70° sin20° + cos20° cosec70° = 2

   Solution :-

   a) sin66° – cos24° = 0

   L.H.S.

   = sin66° – cos24°

   = sin(90° – 24°) – cos24°

   = cos24° – cos24°

   = 0

   ∴ L.H.S. = R.H.S.

   b) cos²57° + cos²33 = 1

   L.H.S.

   = cos²57° + cos²33°

   = cos²(90° – 33°) + cos²33°

   = sin²33° + cos²33°

   = 1

   ∴ L.H.S. = R.H.S.

   c) cos²75° – sin²15° = 0

   L.H.S.

   = cos²75° – sin²15°

   = cos²(90° – 15°) – sin²15°

   = sin²15° – sin²15°

   = 0

   ∴ L.H.S. = R.H.S.

   d) cosec²48° – tan²42° = 1

   L.H.S.

   = cosec²48° – tan²42°

   = cosec²(90° – 42°) – tan²42°

   = sec²42° – tan²42°

   = 1

   ∴ L.H.S. = R.H.S.

   e) sec70° sin20° + cos20° cosec70° = 2

   L.H.S.

   = sec70° sin20° + cos20° cosec70°

   = sec(90° – 20°) sin20° + cos20° cosec(90° – 20°)

   = cosec20° . sin20° + cos20° . sec20°

   = 1 + 1

   = 2

   ∴ L.H.S. = R.H.S.

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2. EXPLANATION.

   (1) = sin66° – cos24° = 1.

   As we know that,

   ⇒ sin(90° – 24°) – cos24°.

   ⇒ cos24° – cos24° = 1.

   (2) = cos²57° + cos²33° = 1.

   As we know that,

   ⇒ cos²57° + cos²(90° – 57°).

   ⇒ cos²57° + sin²57° = 1.

   (3) = cos²75° – sin²15° = 0.

   As we know that,

   ⇒ cos²75° – sin²(90° – 75°).

   ⇒ cos²75° – cos²75° = 0.

   (4) = cosec²48° – tan²42° = 1.

   ⇒ cosec²(90° – 42°) – tan²42°.

   ⇒ sec²42° – tan²42° = 1.

   (5) = sec70°.sin20° + cos20°.cosec70° = 2.

   As we know that,

   ⇒ sec(90° – 20°).sin20° + cos20°.cosec(90° – 20°).

   ⇒ cosec20°/cosec20° + 1/sec20°.sec20°.

   ⇒ 1 + 1 = 2.

   Reply