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Prove that (1-sin^(2)theta)sec^(2)thet...

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. " `
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RS AGGARWAL-TRIGONOMETRIC IDENTITIES-Test Yourself
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  5. If sin theta=(sqrt(3))/(2)" then " ("cosec"theta+cot theta)=?

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  6. If cotA=(4)/(5), then ((sinA+cosA))/((sinA-cosA))=?

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  7. If 2x=secA and (2)/(x)=tanA, " prove that " (x^(2)-(1)/(x^(2)))=(1)/(4...

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  10. If 2sin2 theta=sqrt(3) ,then the value of theta is :

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  11. Prove that sqrt((1+cosA)/(1-cosA))=("cosec"A+cotA).

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  12. If cosec theta + cot theta= p, cos theta = ?

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  13. Prove that ("cosec"A-cotA)^(2)=((1-cosA))/((1+cosA)).

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  14. If 5cottheta=3. find the value of (5sintheta-3costheta)/(4sintheta+3co...

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  15. Prove that (sin32^(@)cos58^(@)+cos32^(@)sin58^(@))=1.

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  16. If x=a sin theta+b cos theta and y=a cos theta -b sin theta, " prove t...

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  17. ((1+sintheta))/((1-sintheta))=?

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  18. Prove that (1)/((sec theta - tan theta))-(1)/(cos theta)=(1)/(cos th...

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  21. If sec5A="cosec"(A-36^(@)) and 5A " is an acute angle, show that " A=2...

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