How do you prove that #sinytany=cosy# is not an identity by showing a counterexample?

Question

4447 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-01-29T10:55:37

It is not an identity. See explanation.

We have

#sinytany=cosy#

This can be written as

#siny*siny/cosy=cosy" "#(#tany=siny/cosy#)

Solving this we get

#sin^2y=cos^2y#

This will be true only when #siny=cosy#.
For example, #sin 45^@=cos4 5^@=1/sqrt2# and so

#sin45^@tan45^@=1/sqrt2xx1=1/sqrt2=cos45^@#

As a counterexample, we can say
#sin30^@tan30^@=1/2xx1/sqrt3=1/(2sqrt3)#

but #cos30^@=sqrt3/2#

Hope that helps!