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Prove the following identities: (ix) ((sinX-sinY)/(cosX+cosY))+((cosX-cosY)/(sinX+sinY))=0 (x) (1/(cosX+sinX-1))+(1/(cosX+sinX+1))=cosecX+secX
Class 10 | Frank | ICSE | Maths | Trigonometric Identities
Prove the following identities: (ix) ((sinX-sinY)/(cosX+cosY))+((cosX-cosY)/(sinX+sinY))=0 (x) (1/(cosX+sinX-1))+(1/(cosX+sinX+1))=cosecX+secX

(ix)

From the question first we consider Left Hand Side (LHS),

=((sinX-sinY)/(cosX+cosY))+((cosX-cosY)/(sinX+sinY))

By taking LCM we get,

=(((sinX+sinY)(sinX-sinY))+((cosX+cosY)(cosX-cosY)))/((cosX+cosY)(sinX-sinY))

By simplification we get,

=\left( \left( {{\sin }^{2}}X-{{\sin }^{2}}Y \right)+\left( {{\cos }^{2}}X-{{\cos }^{2}}Y \right) \right)/\left( \cos X+\cos Y \right)\left( \sin X-\sin Y \right)

=\left( \left( {{\sin }^{2}}X+{{\cos }^{2}}X \right)-\left( {{\sin }^{2}}Y+{{\cos }^{2}}Y \right) \right)/\left( \cos X+\cos Y \right)\left( \sin X-\sin Y \right)

We know that, {{\sin }^{2}}X+{{\cos }^{2}}X=1

=(1-1)/((cosX+cosY)(sinX-sinY))

=0/((cosX+cosY)(sinX-sinY))

=0

Then, Right Hand Side =0

Therefore, LHS = RHS

(x)

From the question first we consider Left Hand Side (LHS),

=(1/(cosX+sinX-1))+(1/(cosX+sinX+1))

By taking LCM we get,

=\left( \cos X+\sin X+1+\cos X+\sin X-1 \right)/\left( {{\left( \cos X+\sin X \right)}^{2}}-1 \right)

We know that, {{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}

=\left( 2\left( \cos X+\sin X \right) \right)/\left( {{\cos }^{2}}X+{{\sin }^{2}}X+2\cos X\sin X-1 \right)

=(cosX+sinX)/(cosXsinX)

=(cosX/(cosXsinX))+(sinX/(cosXsinX))

=(1/sinX)+(1/cosX)

=cosecX+secX

Then, Right Hand Side =cosecX+secX

Therefore, LHS = RHS

i

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