Question

The values of ( tan1° tan2° tan3° . . . . . . . . tan89° ) is

Class 10

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Answer 1

Hey there! Thanks for the question,

We know that tanA = cot(90-A ) and tanA * cotA = 1

So,

( tan1° tan2° tan3° . . . . . . . . tan89° )

= tan1 . tan2 . tan3 ........... tan44 . tan45 . cot(90 - 46 ) .... cot( 90 - 87 ) . cot(90-88 ) . cot(90-89)

= tan1 . tan2 . tan3 ......... tan44 . tan45. cot44 ... cot3 . cot2 . cot1

= ( tan1 . cot1) * ( tan2 . cot2 ) ............ ( tan44. cot44 ) * tan45

= 1 * 1 * 1 *( 44 times) * tan45

= 1 * tan45

= 1

As tan45 = 1 .

Therefore, The value of given expression tan1° tan2° tan3° . . . . . . . . tan89° ) is 1

Answer 2

Given : (tan 1° tan 2° tan 3° ....... tan 89°).

= > tan(90 - 89)tan(90 - 88)tan(90 - 87)....tan(90 - 46)tan 45... tan89

We know that tan(90 - Ф) = cotФ.

= > cot89 * cot88 * cot87...... cot46 * tan 45 * ..... tan 89.

We know that cotФ = (1/tanФ)

= > (1/tan89) * (1/tan 88) * (1/tan 87) ....... (1/tan 46) tan 45 * tan 46 * ...... tan 89

= > tan 45

= > 1.

Hope it helps!