Prove the identity (2*cos 2t / sin 2t) - 2*sin^2t  = cot t + 1

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to prove that (2*cos 2t / sin 2t) - 2*(sin t)^2 = cot t + 1

If this is an identity it should be valid for all values of x for which cos 2t, sin 2t, sin t and cot t are valid.

t = 30 degrees is one such value

(2*cos 2t / sin 2t) - 2*(sin t)^2 = 2* cos 60 / sin 60 - 2* (sin 60)^2

=> 2*2*0.5/sqrt 3 - 2*0.5^2 = 2/sqrt 3 - 0.5

cot 30 + 1 = 3 + 1 = sqrt 3 + 1

As sqrt 3 + 1 is not equal to 2/sqrt 3 - 0.5, the given expression is not an identity.

We can write the left hand side as:

(2*cos 2t / sin 2t) - 2*(sin t)^2

=>2*[(cos t)^2 - (sin t)^2]/2*sin t * cos t - 2*(sin t)^2

=>(cos t/sin t - sin t / cos t - 2*(sin t)^2

=> cot t - tan t - 2*(sin t)^2

The given expression is not an identity.

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