If secx + tan x = p, prove that sin...
Question
If secx + tan x = p, prove that sinx=p2−1b2+1
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If secx + tan x = p, prove that (
sin x=frac{p^{2}-1}{b^{2}+1}
)

JEE/Engineering Exams
Maths
Solution
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If secx + tan x = p, prove that sinx=p2−1b2+1
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( sec x+tan x=p )
( Rightarrow frac{1}{cos x}+frac{sin x}{cos x}=p )
( Rightarrow frac{1+8 tan x}{cos x}=p )
( Rightarrow quad 1+sin x=p cos x )
( Rightarrow(1+sin x)^{2}=left(cos ^{2} xright)^{2} )
( Rightarrow 1+sin ^{2} x+2 sin x=p^{2} cos ^{2} x )
( Rightarrow sin ^{2} x+2 sin x+1=p^{2}left(1-sin ^{2} xright) )
( Rightarrowleft(1+p^{2}right) sin ^{2} x+2 sin x+1-p^{2}=0 )
( Rightarrowleft(1+p^{2}right) sin ^{2} x+left{left(1+p^{2}right)+left(1-p^{2}right)left(sin x+1-p^{2}=0right.right. )
( Rightarrowleft(1+p^{2}right) sin ^{2} x+left(1+p^{2}right) sin x+left(1-p^{2}right) sin x+left(1-p^{2}right)=0 )
( Rightarrowleft(1+p^{2}right) sin x(sin x+1)+left(1-p^{2}right)(sin x+1)=0 )
( Rightarrow(sin x+1)left{left(1+p^{2}right) sin x+left(1-p^{2}right)right}=0 )
( Rightarrow sin x eq 1=0 quad cos left(1+p^{2}right) sin x+left(1-p^{2}right)=0 )
( therefore sin x=frac{p^{2}-1}{p^{2}+1} )

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