Prove following identities, where angles involved are acute angles for which expressions are defined

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8. Introduction to Trigonometry

hard

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

$\left(\frac{1+\tan ^{2} A}{1+\cot ^{2} A}\right)=\left(\frac{1-\tan A}{1-\cot A}\right)^{2}=\tan ^{2} A$

Option A

Option B

Option C

Option D

Solution

$\left(\frac{1+\tan ^{2} A}{1+\cot ^{2} A}\right)=\left(\frac{1-\tan A}{1-\cot A}\right)^{2}=\tan ^{2} A$

$\frac{1+\tan ^{2} A}{1+\cot ^{2} A}=\frac{1+\frac{\sin ^{2} A}{\cos ^{2} A}}{1+\frac{\cos ^{2} A}{\sin ^{2} A}}=\frac{\frac{\cos ^{2} A+\sin ^{2} A}{\cos ^{2} A}}{\frac{\sin ^{2} A+\cos ^{2} A}{\sin ^{2} A}}$

$=\frac{1}{\cos ^{2} A}=\frac{\sin ^{2} A}{\sin ^{2} A}$

$=\tan ^{2} A$

$\left(\frac{1-\tan A }{1-\cot A }\right)^{2}=\frac{1+\tan ^{2} A -2 \tan A }{1+\cot ^{2} A -2 \cot A }$

$=\frac{\sec ^{2} A-2 \tan A}{\operatorname{cosec}^{2} A-2 \cot A}$

$=\frac{\frac{1}{\cos ^{2} A}-\frac{2 \sin A}{\cos A}}{\frac{1}{\sin ^{2} A}-\frac{2 \cos A}{\sin A}}=\frac{\frac{1-2 \sin A \cos A}{\cos ^{2} A}}{\frac{1-2 \sin A \cos A}{\sin ^{2} A}}$

$=\frac{\sin ^{2} A }{\cos ^{2} A }=\tan ^{2} A$

Standard 10

Mathematics

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easy

Prove the following identities, where the angles involved are acute angles for which the expressions are defined. $\left(\frac{1+\tan ^{2} A}{1+\cot ^{2} A}\right)=\left(\frac{1-\tan A}{1-\cot A}\right)^{2}=\tan ^{2} A$

Solution

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Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

$\left(\frac{1+\tan ^{2} A}{1+\cot ^{2} A}\right)=\left(\frac{1-\tan A}{1-\cot A}\right)^{2}=\tan ^{2} A$

State whether the following are true or false. Justify your answer.

$\sin \theta=\cos \theta$ for all values of $\theta$

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$\sqrt{\frac{1+\sin A }{1-\sin A }}=\sec A +\tan A$

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

$\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2 \sec A$

Evaluate the following:

$\frac{5 \cos ^{2} 60^{\circ}+4 \sec ^{2} 30^{\circ}-\tan ^{2} 45^{\circ}}{\sin ^{2} 30^{\circ}+\cos ^{2} 30^{\circ}}$

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The value of $\cos \theta$ increases as $\theta$ increases