Question

The expression (secx + tanx)2 is the same as _____.

Answer 1

The expression
`\bold{(\sec x+\tan x)^(2) \text { is same as } (1+\sin x)/(1-\sin x)}`

Solution:

From question, given that
`\bold{(\sec x+\tan x)^(2)}`

By using the trigonometric identity
`(a + b)^(2) = a^(2) + 2ab + b^(2)` the above equation becomes,

`(\sec x+\tan x)^(2) = \sec ^(2) x+2 \sec x \tan x+\tan ^(2) x`

We know that
`\sec x=(1)/(\cos x) ; \tan x=(\sin x)/(\cos x)`

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

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

On simplication we get

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

By using the trigonometric identity
`\cos ^(2) x=1-\sin ^(2) x` ,the above equation becomes

`=(1+2 \sin x+\sin ^(2) x)/(1-\sin ^(2) x)`

By using the trigonometric identity
`(a+b)^(2)=a^(2)+2ab+b^(2)`

we get
`1+2 \sin x+\sin ^(2) x=(1+\sin x)^(2)`

`=((1+\sin x)^(2))/(1-\sin ^(2) x)`

`=((1+\sin x)(1+\sin x))/(1-\sin ^(2) x)`

By using the trigonometric identity
`a^(2)-b^(2)=(a+b)(a-b)` we get
`1-\sin ^(2) x=(1+\sin x)(1-\sin x)`

`=((1+\sin x)(1+\sin x))/((1+\sin x)(1-\sin x))`

`= (1+\sin x)/(1-\sin x)`

Hence the expression
`\bold{(\sec x+\tan x)^(2) \text { is same as } (1+\sin x)/(1-\sin x)}`