Prove these 4 trigonometric identitites Indicate clearly wh

- Prove these 4 trigonometric identitites. Indicate clearly which identity you are proving.

- explain every single step of the process of proving the identity.

Solution

1)[(1+cosx)_/sinx]+[sinx_/(1+cosx)]

=[(1+cosx)²+(sinx)²]_{/[sinx(1+cosx)]}

=[1+2cosx+cos²x+sin²x]_{/[sinx(1+cosx)]}

=[1+2cosx+1]_{/[sinx(1+cosx)]}.......................................[identity used is cos²x+sin²x =1]

=[2+2cosx]_{/[sinx(1+cosx)]}

=[2(1+cosx)]_{/[sinx(1+cosx)]}

=2_/sinx

=2cscx----------------------------------->(identity used is 1/sinx =cscx)

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cotx -tanx

=(cosx_/sinx)-(sinx_/cosx)------------->(identity used is cotx =cosx/sinx , tanx =sinx/cosx)

=(cos²x-sin²x)_/(sinxcosx)

=(cos²x-(1-cos²x))_/(sinxcosx)......................................[identity used is cos²x+sin²x =1=>sin²x =1-cos²x]

=(cos²x-1+cos²x)_/(sinxcosx)

=(2cos²x-1)_/(sinxcosx)

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(sin²+2sin+1)_/cos²

=(sin +1)²_/cos²...............................(identity used is a²+2ab+b²=1)

=(sin +1)²_/(1-sin²)......................................[identity used is cos²x+sin²x =1 =>1-sin²x=cos²x]

=(1+sin)²_{/(1+sin)(1-sin)}............................(identity used is a²-b²=(a+b)(a-b))

=(1+sin)_/(1-sin)

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(sin²_/cos)+cos

=(sin²+cos²)_/cos..................................[identity used is sin²x+cos²x =1]

=1_/cos

=sec