Question 6.6.5.8: Establishing an Identity Prove the identity: tan (θ + π/2) =......
Establishing an Identity
Prove the identity: \tan\!\left(\theta+{\frac{\pi}{2}}\right)=-\cot\theta
Formula (6) cannot be used because \tan \frac{\pi}{2} is not defined. Instead, proceed as follows:
\tan\left(\alpha+\beta\right)\,={\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}} (6)
\tan\left(\theta+{\frac{\pi}{2}}\right)={\frac{\sin\left(\theta+{\frac{\pi}{2}}\right)}{\cos\left(\theta+{\frac{\pi}{2}}\right)}}={\frac{\sin\theta\cos{\frac{\pi}{2}}+\cos\theta\sin{\frac{\pi}{2}}}{\cos\theta\cos{\frac{\pi}{2}}-\sin\theta\sin{\frac{\pi}{2}}}}={\frac{(\sin\theta)\left(0\right)\,+\,(\cos\theta)\left(1\right)}{(\cos\theta)\left(0\right)\,-\,\left(\sin\theta\right)\left(1\right)}}={\frac{\cos\theta}{-\sin\theta}}=-\cot \theta
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