Verify the following identities a tantheta cottheta sec th

Verify the following identities. a. tan(theta) + cot(theta) = sec (theta) csc (theta) b. 1/1 + sin(theta) + 1/1 - sin(theta) = 2 sec^2(theta) c. sec(theta)/1 + sec(theta) = 1/1 + cos(theta)

Solution

(a)

tan() +cot()

=(sin()_/cos())+(cos()_/sin())

=(sin²()+cos²()_{/(cos()sin())}

=1_{/(cos()sin())}

=(1_/cos())*(1_/sin())

=sec()csc()

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(b)

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

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

=2_{/((1+sin())(1-sin()))}

=2_/(1-sin²())

=2_/cos²()...........................................................[since sin²()+cos²()=1 is an identity]

=2sec²()

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(c)

sec()_/(1+sec())

=1_{/(cos()*(1+sec()))}

=1_{/(cos() +(cos()sec()))}

=1_{/(cos() +1)}

=1_/(1+cos())