Question

Prove that

(1 - sin²theta) sec²theta = 1

(1 + Tan ²theta)cos²theta = 1

$2cos {}^{2} theta \: + \frac{2}{(1 + cot {}^{2}theta } ) = 2$
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Answer 1

✬ LHS = RHS ✬

Step-by-step explanation:

Given:

• (1 – sin²θ)sec²θ = 1
• (1 + tan²θ)cos²θ = 1
• 2cos²θ + 2/(1 + cot²θ) = 2

To Prove:

• LHS = RHS

Proof: We will use several formulae here

• (sin²θ + cos²θ) = 1

• sec²θ = 1/cos²θ

• 1 + tan²θ = sec²θ

• 1 + cot²θ = cosec²θ

• 1/cosec²θ = sin²θ

[ First ]

$\implies{\rm }$ (1 – sin²θ)sec²θ = 1

$\implies{\rm }$ (1 – sin²θ) × 1/cos²θ = 1

$\implies{\rm }$ cos²θ × 1/cos²θ = 1

$\implies{\rm }$ 1 = 1

$\large\purple{\texttt { Proved}}$

________________________

[ Second ]

$\implies{\rm }$ (1 + tan²θ)cos²θ = 1

$\implies{\rm }$ sec²θ × 1/sec²θ = 1

$\implies{\rm }$ 1 = 1

$\large\orange{\texttt { Proved}}$

________________________

[ Third ]

$\implies{\rm }$ 2cos²θ + 2/(1 + cot²θ) = 2

$\implies{\rm }$ 2cos²θ + 2/cosec²θ = 2

$\implies{\rm }$ 2cos²θ + 2 × 1/cosec²θ = 2

$\implies{\rm }$ 2cos²θ + 2 × sin²θ = 2

$\implies{\rm }$ 2(cos²θ + sin²θ) = 2

$\implies{\rm }$ 2 × 1 = 2

$\implies{\rm }$ 2 = 2

$\large\red{\texttt { Proved}}$

Answer 2

AnswEr :-

1]

LHS

$\sf (1-\sin^2\theta) \times sec^2\theta$

$\sf (1-\sin^2\theta)\times \dfrac{1}{cos\theta} \bigg(sec^2\theta = \dfrac{1}{cos^2\theta}\bigg)$

$\sf 1 - \sin^2\theta-\dfrac{1}{cos\theta}$

$\sf cos^2\theta \times \dfrac{1}{cos\theta}\bigg(1-sin^2= cos^2\bigg)$

$\sf 1$

RHS

$1$

2]

LHS

$\sf (1+tan^2\theta)\times cos^2\theta$

$\sf 1 + tan^2\theta\times cos^2\theta$

$\sf sec^2\theta\times cos^2\theta\bigg(1 + tan^2 = sec^2\bigg)$

$\sf sec^2 \theta \times \dfrac{1}{sec^2\theta}\bigg(cos^2 = \dfrac{1}{sec^2}\bigg)$

$\sf 1$

RHS

$\sf 1$

3]

LHS

$\sf 2cos^2\theta + \bigg(\dfrac{2}{1+cos^2\theta}\bigg)$

$\sf 2cos^2\theta +\bigg( \dfrac{2}{cosec^2\theta}\bigg)\bigg(1 + cot^2=cosec^2\bigg)$

$\sf 2cos^2\theta+\bigg(2\times\dfrac{1}{cosec^2\theta}\bigg)$

$\sf 2cos^2\theta+2sin^2\theta\bigg(\dfrac{1}{cosec^2}=sin^2\bigg)$

$\sf 2(cos^2 \theta+sin^2\theta)$

$\sf 2(1) \bigg(cos^2 + sin^2=1\bigg)$

$\sf 2$

RHS

$\sf 2$