Question

Please simplify the following trigonometric identity.
$\displaystyle\frac{1}{\sec\alpha-\tan\alpha}​$​

Answer 1

Answer:

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Answer 2

To Find :

simplify the following trigonometric identity :- $\displaystyle\frac{1}{\sec\alpha-\tan\alpha}$

Solution :

$\implies \sf \dfrac{1}{ \sec\alpha-\tan\alpha}$

$\implies \sf \dfrac{1}{\sec\alpha}-\dfrac{1}{\tan\alpha}$

$\implies \sf \dfrac{1}{\dfrac{1}{\cos\alpha}}-\dfrac{1}{\dfrac{1}{\tan\alpha}}$

$\implies\sf \dfrac{\cos\alpha}{1}-\dfrac{\cos\alpha}{\sin\alpha}$

$\implies\sf \dfrac{\sin\alpha\cos\alpha-\cos\alpha}{\sin\alpha}$

$\implies \sf \cos\alpha \Big(\dfrac{\sin\alpha-1}{\sin\alpha}\Big)$

$\implies\sf \dfrac{ \cos\alpha}{\sin\alpha} \Big(\sin\alpha-1 \Big)$

$\implies\sf \cot\alpha \Big(\sin\alpha-1 \Big)$

Formulas Applied :

• $\displaystyle\sf \frac{1}{\sec\theta}=\cos\theta$

• $\displaystyle\sf \tan\theta=\frac{\sin\theta}{\cos\theta}$

• $\displaystyle \sf \frac{\cos\theta}{\sin\theta}=\cot\theta$

Required Answer :

${\boxed{\boxed{\displaystyle\frac{1}{\sec\alpha-\tan\alpha}=\cot\alpha \Big(\sin\alpha-1 \Big)}}}$