Question

Verify the identity (tan x + sec x)(sin x+ 1)= cos x ?

Answer 1

`\mleft(\tan x+secx\mright)\mleft(1-\sin x\mright)=cosx`
`Explanation\colon`
`\begin{gathered} \tan x=(\sin x)/(\cos x) \\ \sec x=(1)/(\cos x) \\ Startthesimplificationprocess\colon \end{gathered}`
`\begin{gathered} \mleft((\sin x)/(\cos x)+(1)/(\cos x)\mright)(1-\sin x) \\ ((\sin x+1))/(\cos x)(1-\sin x) \end{gathered}`
`\begin{gathered} (1-\sin ^2x)/(\cos x) \\ (\cos ^2x)/(\cos x) \end{gathered}`

Now, rearrange the pythagorean identity

`\begin{gathered} \sin ^2x+\cos ^2x=1 \\ \cos ^2x=1-\sin ^2x \end{gathered}`
`\begin{gathered} ((\cos x)(\cos x))/(\cos x) \\ =\cos x \end{gathered}`

`\mleft(\tan x+secx\mright)\mleft(1-\sin x\mright)=cosx`

PROOF