Question

Cos^2x+4cosx+4/cosx+2=2secx+1/secx

Answer 1

`(\cos^2x+4\cos x+4)/(\cos x+2)=(2\sec x+1)/(\sec x)`

Explanation:

To prove the given identity, we solve the left hand side and right hand side expressions and show that they are equal.

So we get
`(2\sec x+1)/(\sec x)\\\\=(2\sec x)/(\sec x)+(1)/(\sec x)\\\\= 2+\cos x\\\\\text{since the left hand side and right hand side expression are same, so}\\\text{it verifies the identity.}`

`\text{Left hand side}=(\cos^2x+4\cos x+4)/(\cos x+2)\\\\\text{Here observe that the numerator is a perfect square using }\\a^2+2ab+b^2=(a+b)^2\\\text{so we get}\\\\=((\cos x+2)^2)/(\cos x+2)\\\\\text{now we can cancel out one factor (cos x+2) from numerator and denominator.}\\\text{so we get}\\\\=\cos x+2\\\\\text{And similarly the right hand side gives}`