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Prove the following identities: (i) {{\left( 1+\cot B \right)}^{2}}+{{\left( 1-\cot B \right)}^{2}}=2\cos e{{c}^{2}}B (ii) (cosecA/(tanA+cotA))=cosA
Class 10 | Frank | ICSE | Maths | Trigonometric Identities
Prove the following identities: (i) {{\left( 1+\cot B \right)}^{2}}+{{\left( 1-\cot B \right)}^{2}}=2\cos e{{c}^{2}}B (ii) (cosecA/(tanA+cotA))=cosA

(i)

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

={{\left( 1+\cot B \right)}^{2}}+{{\left( 1-\cot B \right)}^{2}}

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

=1+{{\cot }^{2}}B+2\cot B+1+{{\cot }^{2}}B-2\cot B

=2+2{{\cot }^{2}}B

Taking common terms outside we get,

=2\left( 1+{{\cot }^{2}}B \right)

Also we know that, 1+{{\cot }^{2}}B=\cos e{{c}^{2}}B

=2\cos e{{c}^{2}}B

Then, Right Hand Side=2\cos e{{c}^{2}}B

Therefore, LHS = RHS

(ii)

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

=(cosecA/(tanA+cotA))

We know that, cosecA=1/sinA, tanA=sinA/cosA, cotA=cosA/sinA

Then,

=(1/sinA)/((sinA/cosA)+(cosA/sinA))

Taking LCM in the denominator we get,

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

Also we know that, {{\sin }^{2}}A+{{\cos }^{2}}A=1

=(1/sinA)/(1/cosAsinA)

=\left( 1/\sin A \right)\times \left( \left( \cos A\sin A \right)/1 \right)

=cosA

Then, Right Hand Side =cosA

Therefore, LHS = RHS

i

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