1.

If A + B = \(\frac{\pi}{4},\)then prove that (1 + tan A) (1 + tan B) = 2.

Answer»

Given,

A + B = \(\frac{\pi}{4}\)

Taking tangent both sides, we get

tan(A + B) = tan\(\frac{\pi}{4}\)

or, \(\frac{tan\,A+tan\,B}{1-tan\,A\,tan\,B}=1\)

or, tan A + tan B = 1 – tan A tan B

or, tan A + tan B + tan A tan B = 1

Now, adding 1 both sides, we get

tan A + tan B + tan A tan B + 1 = 2

(tan A + 1) + tan B (1 + tan A) = 2

or, (tan A + 1) (1 + tan B) = 2

or, (1 + tan A) (1 + tan B) = 2



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