| Assertion (A): x\[{{\cos }^{2}}A-{{\sin }^{2}}A=1,\] \[{{\tan }^{2}}A-{{\sec }^{2}}A=1\] are trigonometric identities. |
| Reason (R): An equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true for all values of the angles involved. |
A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A)
B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A)
C) Assertion (A) is true but Reason (R) is false
D) Assertion (A) is false but Reason (R) is true
Correct Answer: D
Solution :
| [d] We have \[{{\cos }^{2}}A-{{\sin }^{2}}A=1\] |
| Put \[A={{45}^{o}}\] we get |
| \[{{\cos }^{2}}{{45}^{\operatorname{o}}}-{{\sin }^{2}}{{45}^{\operatorname{o}}}={{\left( \frac{1}{\sqrt{2}} \right)}^{2}}-{{\left( \frac{1}{\sqrt{2}} \right)}^{2}}=0\ne 1\] |
| and \[{{\tan }^{2}}A-{{\sec }^{2}}A=1\] |
| Put \[A={{45}^{\operatorname{o}}}\] we get, |
| \[{{\tan }^{2}}{{45}^{\operatorname{o}}}-{{\sec }^{2}}{{45}^{\operatorname{o}}}=1-{{(\sqrt{2})}^{2}}=-1\ne 1\] |
| \[\therefore \] These are not trigonometric identities. |
| \[\therefore \] Assertion False ; Reason : True. |
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