Question

prove that secθ - cosθ = tan θ sinθ​

Answer 1

Answer:

LHS

secθ - cosθ

= 1/cosθ - cosθ

= (1-cos²θ)/cosθ

= sin²θ/cosθ

= sinθ/ cosθ x sinθ

= tanθ sinθ

= RHS. Hence proved

Hope this helps ☺️

Answer 2

ANSWER:

To Prove:

• secθ - cosθ = tanθ.sinθ

Proof:

$\text{We need to prove that,}\\\\:\longrightarrow\sec\theta-\cos\theta=\tan\theta.\sin\theta\\\\\text{On solving LHS,}\\\\:\implies\sec\theta-\cos\theta\\\\\text{We know that,}\\\\:\hookrightarrow\sec\theta=\dfrac{1}{\cos\theta}\\\\\text{So,}\\\\:\implies\sec\theta-\cos\theta\\\\:\implies\dfrac{1}{\cos\theta}-\cos\theta\\\\\text{On taking LCM,}\\\\:\implies\dfrac{1}{\cos\theta}-\cos\theta\\\\:\implies\dfrac{1-\cos^2\theta}{\cos\theta}$

$\text{We know that,}\\\\:\hookrightarrow1-\cos^2\theta=\sin^2\theta\\\\\text{So,}\\\\:\implies\dfrac{1-\cos^2\theta}{\cos\theta}\\\\:\implies\dfrac{\sin^2\theta}{\cos\theta}\\\\:\implies\dfrac{\sin\theta\times\sin\theta}{\cos\theta}\\\\:\implies\dfrac{\sin\theta}{\cos\theta}\times\sin\theta\\\\\text{We know that,}\\\\:\hookrightarrow\dfrac{\sin\theta}{\cos\theta}=\tan\theta\\\\\text{So,}\\\\:\implies\dfrac{\sin\theta}{\cos\theta}\times\sin\theta\\\\\bf{:\implies\tan\theta\times\sin\theta=RHS}\\\\\text{\bf{HENCE PROVED!!!}}$

Formulae Used:

$1.\tan\theta=\dfrac{\sin\theta}{\cos\theta}\\\\2.1-\cos^2\theta=\sin^2\theta\\\\3.\sec\theta=\dfrac{1}{\cos\theta}$