Question

Prove that: cos ​2 theta/1-tan theta + sin 2 theta/1-cot theta = 1+sin theta cos theta

Answer 1

Step-by-step explanation:

LHS

$\frac{ {cos}^{2} \theta}{1 - tan \theta} + \frac{ {sin}^{2} \theta }{1 - cot \theta} \\ \\ = \frac{ {cos}^{3} \theta }{cos \theta - sin \theta} + \frac{ {sin}^{3} \theta }{sin \theta - cos \theta}$

$= \frac{ {cos}^{3} \theta}{cos \theta - sin \theta} - \frac{ {sin}^{3} \theta}{cos \theta - sin \theta} \\ \\ = \frac{ {cos}^{3} \theta - {sin}^{3} \theta}{cos \theta - sin \theta}$

$= \frac{(cos \theta - sin \theta)(1 +c os \theta sin \theta )}{cos \theta - sin \theta} \\ \\ = 1 + sin \theta cos \theta$

RHS

Answer 2

Answer:

1+sin theta×cos theta

Step-by-step explanation:

Step 1: The area of mathematics known as trigonometry examines the link between the ratios of a right-angled triangle's sides to its angles. Trigonometric ratios, such as sine, cosine, tangent, cotangent, secant, and cosecant, are employed to analyse this connection. The notion of trigonometry was developed by the Greek mathematician Hipparchus, while the name trigonometry is a 16th century Latin derivation.

cos2theta/(1-tan theta)  +  sin2theta/(1-cot theta)

Step 2: change tanthetha and cotthetha into sin and cos.

=> cos^thetha/(costhetha – sinthetha) – sin^3thetha/(sinthetha -costhetha)

=> (cos^3thetha – sin^3thetha)/(costhetha -sinthetha)

=> (costhetha-sinthetha)(1 + sinthethacosthetha)/ (costhetha-sinthetha)

=>  1+sin theta×cos theta

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