Question

prove the following :
$(1 \div 1 + \sin^{2} ) + (1 \div 1 + \cos^{2} ) + (1 \div 1 + \sec^{2} ) + (1 \div 1 + csc^{2} ) = 2$
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Answer 1

Solution:

i) 1/(1+sec²A)

= 1/(1+1/cos²A)

= cos²A/(cos²A+1) ---(1)

ii) 1/(1+cosec²A)

= 1/(1+1/sin²A)

= sin²A/(sin²A+1) ------(2)

Now,

LHS = prove the following :

$(1 \div 1 + \sin^{2} ) + (1 \div 1 + \cos^{2} ) + (1 \div 1 + \sec^{2} ) + (1 \div 1 + csc^{2} )$

= 1/(1+sin²A)+1/(1+cos²A)

+ cos²A/(1+cos²A)+sin²A/(1+sin²A)

{ from (1) and (2) }

=[1/(1+sin²A) + sin²A/(1+sin²A)]

+[1/(1+cos²A)+cos²A/(1+cos²A)]

= [(1+sin²A)/(1+sin²A)]+

[(1+cos²A)/(1+cos²A)]

After cancellation, we get

= 1+1

= 2

= RHS