Question

Show that : tan 10° tan 15° tan 75° tan 80° = 1

Answer 1

tan 10° tan 15° tan 75° tan 80° = 1

We will expand the said values -

= tan 10° tan 15° tan ( 90° - 15°) tan(90° - 10° )

= tan 10° tan 15° cot 15° cot 10° [ Since tan ( 90°-Ф = cot Ф)

Thus after substituting the value -

= tan 10° × 1 / tan 10° × tan 15° × 1 / tan 15°

= 1 × 1

= 1

Answer 2

Step-by-step explanation:

L.H.S. = tan 10˚ tan 15˚ tan 75˚ tan 80˚

= tan 10˚ tan 15˚ tan (90˚ – 15˚) tan(90˚ – 10˚)

= tan 10˚ tan 15˚ cot 15˚ cot 10˚

1/cot10˚ × 1/cot 15˚ × cot 15˚ × cot 10˚

= 1 = R.H.S.

Hence proved.

(ii) L.H.S. = cos 1˚ cos 2˚ cos 3˚ …..cos 180˚

= cos1˚ cos2˚ cos 3˚ …..cos 89˚ cos 90˚ ….cos 180˚

= cos 1˚ cos 2˚ cos 3˚ ….cos 89˚ × 0 × cos 91˚ ….cos 180˚

= 0 = R.H.S.

Hence proved.