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# Prove the following identities - CBSE Class 10 - Mathematics

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#### Question

Prove the following identities:

( i)sin^{2}A/cos^{2}A+\cos^{2}A/sin^{2}A=\frac{1}{sin^{2}Acos^{2}A)-2

(ii)\frac{cosA}{1tanA}+\sin^{2}A/(sinAcosA)=\sin A\text{}+\cos A

( iii)((1+sin\theta )^{2}+(1sin\theta)^{2})/cos^{2}\theta =2( \frac{1+sin^{2}\theta}{1-sin^{2}\theta } )

#### Solution

(i)We have,

LHS=sin^{2}A/cos^{2}A+cos^{2}A/sin^{2}A =\frac{sin^{4}A+cos^2A}{sin^{2}Acos^{2}A}

=((\sin ^{2}A)^{2}+(\cos ^{2}A)^{2}+2\sin^{2}A\cos ^{2}A-2\sin^{2}Acos^{2}A)/(\sin^{2}A\cos ^{2}A)

=((\sin ^{2}A+\cos ^{2}A)^{2}-2\sin ^{2}A\cos^{2}A)/(sin ^{2}A\cos ^{2}A

=(1-2\sin ^{2}A\cos ^{2}A)/(\sin^{2}A\cos ^{2}A)

=1/(\sin^{2}A\cos ^{2}A)-2=RHS

(ii) We have,

LHS=\frac{\cos A}{1-\tan A}+\sin ^{2}A/(\sin A-\cos A)

=\frac{\cos A}{1-\frac{\sin A}{\cos A}}+\sin^{2}A/(\sinA-\cos A)

=\frac{\cos A}{\frac{\cos A-\sin A}{\cos A}}+\sin ^{2}A/(\sinA-\cos A)

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

=\cos ^{2}A/(\cos A\sin A)-\sin ^{2}A/(\cos A\sin A)

=(\cos ^{2}A-\sin ^{2}A)/{\cos A-\sin A}

=\frac{(\cos A+\sin A)\,(\cos A-\sin A)}{\cos A-\sin A}

= cos A + sin A = RHS

(iii) We have,

LHS=((1+\sin \theta )^{2}+(1\sin \theta )^{2})/(\cos^{2}\theta )

=\frac{(1+2\sin \theta +\sin ^{2}\theta )+(12\sin \theta +\sin^{2}\theta )}{\cos ^{2}theta

=(2+2\sin^{2}\theta)/ \cos ^{2}\theta=(2(1+\sin^{2}\theta ))/{1-\sin ^{2}\theta }=2( \frac{1+\sin^{2}\theta }{1-\sin ^{2}\theta } )

= RHS.

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Solution Prove the following identities Concept: Trigonometric Identities.
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