Question

e. Prove that
1+sing
1-sing
+
1-sing
=2 sec
1+sin​

Answer 1

Answer:Let  y=1+sinx−−−−−−−√1−sinx−−−−−−−√+1−sinx−−−−−−−√1+sinx−−−−−−−√  

⟹y=(1+sinx)2−−−−−−−−−√+(1−sinx)2−−−−−−−−−√1−sinx−−−−−−−√1+sinx−−−−−−−√  

⟹y=1+sinx+1−sinx1−sinx−−−−−−−√1+sinx−−−−−−−√  

⟹y=21−sin2x−−−−−−−−√  

⟹y=2cos2x−−−−−√  

⟹y=2cosx  

⟹y=2secx

Step-by-step explanation:√(1+ sin x)/√(1- sin x) +√(1-sin x)/√(1+sin x)=2sec x

LHS = √(1+ sin 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)/(1+sin x)

= √(1+ sin x)√(1- sin x)[1/(1-sin x)+ 1/(1+sin x)

= √(1- sin^2 x)[{1+sin x+1-sin x}/(1-sin^2 x)

= √cos^2 x)[2/(1-sin^2 x)

= 2 cos x/cos^2 x

= 2/cos x

= 2 sec x = RHS.