Solve the following
$CosA-SinA+\frac{1}{CosA}+SinA-1=CosecA+CotA$

Given: $CosA-SinA+\frac{1}{CosA}+SinA-1=CosecA+CotA$

To do: Prove LHS=RHS

Solution:

LHS = $\frac{CosA-SinA+1}{CosA+SinA-1}$

Dividing numerator and denominator by SinA

= $\frac{ \frac{CosA}{SinA} - \frac{SinA}{SinA} + \frac{1}{SinA}} {\frac{CosA}{SinA} + \frac{SinA}{SinA} - \frac{1}{SinA}}$

= $\frac{CotA - 1 + CosecA}{CotA + 1 - CosecA}$

=$\frac{(CotA + CosecA) - (CosecA)^{2} - (CotA)^{2}}{(CotA + 1 - CosecA)}$  

Since , $(CosecA)^{2} - (CotA)^{2}= 1$

= $\frac{(CotA + CosecA)-(CotA + CosecA)(- CotA + CosecA)} {CotA + 1 - CosecA}$

= $\frac{(CotA + CosecA)((CotA + 1 - CosecA)}{CotA + 1 - CosecA}$

= $(CosecA + CotA)$ = RHS

Tutorialspoint

Updated on: 10-Oct-2022

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