Verify this triganomic identity you can only use 1 side to solve (cosx-tanx)/(sinx+cosx)=cosx-secx

Verify this triganomic identity you can only use 1 side to solve (cosx-tanx)/(sinx+cosx)=cosx-secx

Question
Verify this triganomic identity you can only use 1 side to solve \(\displaystyle\frac{{{\cos{{x}}}-{\tan{{x}}}}}{{{\sin{{x}}}+{\cos{{x}}}}}={\cos{{x}}}-{\sec{{x}}}\)

Answers (1)

2021-01-20
First we need to check the identity. Let x=0 then:
\(\displaystyle\frac{{{1}-{0}}}{{{0}+{1}}}={1}-{1}\), which isn’t true because \(\displaystyle{1}\ne{0}\), so the identity is false.
To confirm, let \(\displaystyle{x}=\frac{\pi}{{4}}:\)
\(\displaystyle\frac{{\frac{\sqrt{{2}}}{{2}}-{1}}}{{s}}{q}{r}{2}={s}{q}{r}\frac{{2}}{{2}}-{s}{q}{r}{2},\frac{{{1}-{s}{q}{r}{2}}}{{2}}\ne-{s}{q}{r}\frac{{2}}{{2}}.\)
Therefore the identity has been wrongly stated. So the identity is actually an equation which has particular solutions for x. An identity has to be true for all x.
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