# Prove that  (1-sin^(2)theta)sec^(2)theta=1

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Prove that  (1-sin^(2)theta)sec^(2)theta=1

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We have
(i)" LHS "= (1-sin^(2)theta)sec^(2) theta
 = cos^(2)theta* sec^(2) theta " "[because 1-sin^(2)theta=cos^(2)theta ]
 = cos^(2)theta * (1)/(cos^(2)theta)=1= " RHS."
 therefore " LHS " = " RHS. "
 (ii) " LHS " = (1+ tan^(2) theta )cos^(2) theta
= sec^(2)theta* cos^(2)theta " "[because 1+ tan^(2)theta=sec^(2)theta ]
 = (1)/(cos^(2)theta)*cos^(2)theta=1="RHS."
 therefore " LHS " = " RHS."
 (iii) " LHS "= (1+ tan^(2)theta)(1-sin theta)(1+sin theta)
 = (1+tan^(2) theta)(1-sin^(2)theta)= sec^(2) theta * cos^(2) theta
 [ because (1+ tan^(2) theta )= sec^(2)theta , (1- sin^(2)theta )= cos^(2)theta]
 =(1)/(cos^(2) theta)* cos^(2)theta " "[ because sec^(2) theta=(1)/(cos^(2)theta)]
 =1= " RHS. "
 therefore " LHS " = " RHS. "
 (iv) " LHS "= 2 cos^(2) theta+(2)/((1+cot^(2) theta))
 =2 cos^(2) theta+(2)/("cosec"^(2)theta )" "[ because 1+cot^(2)theta="cosec"^(2) theta]
=2 cos^(2)theta+ 2sin^(2) theta " "[ because (1)/("cosec"^(2) theta)=sin^(2) theta]
=2(cos^(2)theta + sin^(2) theta)= 2 xx 1 " "[ because cos^(2) theta +sin^(2) theta=1 ]
= 2= "RHS."
 therefore " LHS " = " RHS. "