Question

prove that
$cos^2(1 + tan ^2) = 1$
​

Answer 1

Given:-

• $\tt{cos^2\theta(1+tan^2\theta)=1}$

To Prove:-

• $\tt{LHS = RHS} .$

Formulae Used:-

• $\tt{\red{sin^2\theta+cos^2\theta=1}}$
• $\tt{\red{\tan^2=\dfrac{\sin^2\theta}{\cos^2\theta}}}$

Proof:-

Given that cos²θ ( 1 + tan²θ) = 1.

Let us simply LHS and try to obtain RHS :

LHS

= cos²θ ( 1 + tan²θ ) .

= cos²θ ( 1 + sin²θ/cos²θ ).

= cos²θ ( cos²θ + sin²θ / cos²θ ).

= cos²θ ( sin²θ + cos²θ / cos²θ).

= cos²θ × 1/cos²θ .

= 1.

= RHS .

Hence Proved : )

More to know:-

Here are some more identities :-

$\bullet$ sin²θ + cos²θ = 1.

$\bullet$ sec²θ -tan²θ = 1.

$\bullet$ cosec²θ - cot²θ = 1.

Trigonometric Ratios of Compound Angles:-

$\bullet$sin(A+B)=sinA.cosB+cosA.sinB

$\bullet$sin(A-B)=sinA.cosB-cosA.sinB

$\bullet$ cos(A-B)=cosA.cosB+sinA.sinB

$\bullet$ cos(A+B)=cosA.cosB-sinA.sinB

$\bullet$ tan(A+B)=tanA+tanB/1-tanA.tanB

$\bullet$ tan(A-B)=tanA-tanB/1+tanA.tanB