Answer:
[tex]\frac{\cos^2x+4\cos x+4}{\cos x+2}=\frac{2\sec x+1}{\sec x}[/tex]
Step-by-step 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[tex]\frac{2\sec x+1}{\sec x}\\\\=\frac{2\sec x}{\sec x}+\frac{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.}[/tex]
[tex]\text{Left hand side}=\frac{\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}\\\\=\frac{(\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}[/tex]
