Question

Prove the following trigonometric identities:
tanθ+1/tanθ=secθcosecθ

Answer 1

Answer:

hey mate

Step-by-step explanation:

LHS = tanθ/(1 - cotθ) + cotθ/(1 - tanθ)

= tanθ/(1 - 1/tanθ) + (1/tanθ)/(1 - tanθ)

= tan²θ/(tanθ - 1) + 1/tanθ(1 - tanθ)

= tan³θ/(tanθ - 1) - 1/tanθ(tanθ - 1)

= (tan³θ - 1)/tanθ(tanθ - 1)

= (tanθ - 1)(tan²θ + 1 + tanθ)/tanθ(tanθ - 1)

= (tan²θ + 1 + tanθ)/tanθ

= tanθ + cotθ + 1

= sinθ/cosθ + cosθ/sinθ + 1

= (sin²θ + cos²θ)/sinθ.cosθ + 1

= secθ.cosecθ + 1

= 1 + secθ.cosecθ = RHS

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Answer 2

Proving:

Step-by-step explanation:

$\ Given \ value: \\\\\tan\theta \ +\frac{1}{\tan \theta} \ = \sec \theta \ cosec \theta \\\\\ Solution: \\\\\ L.H.S \\\\ \rightarrow \tan\theta \ +\frac{1}{\tan \theta} \\\\ \rightarrow \frac{\tan^2\theta \ + \ 1}{\tan \theta} \\\\\ formula: \\\\\sec^2 \theta \ - \tan^2 \theta = \ 1 \\\\\sec^2 \theta \ =\ 1\ +\tan^2 \theta \ \ \ \ \ \ \ and \ \ \ \ \ \tan\theta \ = \frac{sin\theta}{\cos\theta}\\\\ \rightarrow \frac{\sec^2 \theta }{\tan \theta}$

$\rightarrow \frac{\sec \theta(\cos\theta)}{\sin \theta(\cos\theta)} \\\\\rightarrow \frac{\sec \theta}{\sin \theta} \\\\\rightarrow \sec \theta\ cosec \ \theta}$

L.H.S = R.H.S

Learn more:

• Proving: https://brainly.in/question/8112701