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

1 Answer
Jan 29, 2017

It is not an identity. See explanation.

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!