Solve the equation `5sin^(2)x-7sinx cosx+16cos^(2)x=4`

Explanation

To solve this equation we use the fundamental formula of trignometric identities,
`sin^(2)x + cos^(2)x=1`
writing the equation in the form,
`5sin^(2)x-7sinx.cosx+16cos^(2)x=4(sin^(2)x+cos^(3)x)`
`rArr sin^(2)x-7sinxcosx+12cos^(2)x=0`
Dividing by `cos^(2)x` on both side we get
`tan^(2)x-7tanx+12=0`
Now it can be factorized as:
`(tanx-3)(tanx-4)=0`
`rArr tanx=3,4`
i.e., `tanx=tan(tan^(-1)3)` or `tanx=tan(tan^(-1)4)`
`rArr x=npi+tan^(-1)3` or `x=npi+tan^(-1)4, n in I`.Ans.