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Trigonometry Calculator

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1

Here, we show you a step-by-step solved example of trigonometry. This solution was automatically generated by our smart calculator:

$\frac{1+cscx}{secx}-cotx=cosx$
2

Starting from the left-hand side (LHS) of the identity

$\frac{1+\csc\left(x\right)}{\sec\left(x\right)}-\cot\left(x\right)$
3

Combine all terms into a single fraction with $\sec\left(x\right)$ as common denominator

$\frac{1+\csc\left(x\right)-\cot\left(x\right)\sec\left(x\right)}{\sec\left(x\right)}$

Applying the trigonometric identity: $\cot\left(\theta \right) = \frac{\cos\left(\theta \right)}{\sin\left(\theta \right)}$

$\frac{1+\csc\left(x\right)+\frac{-\cos\left(x\right)}{\sin\left(x\right)}\sec\left(x\right)}{\sec\left(x\right)}$

Applying the secant identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$

$\frac{1+\csc\left(x\right)+\frac{-\cos\left(x\right)}{\sin\left(x\right)}\frac{1}{\cos\left(x\right)}}{\sec\left(x\right)}$

Multiplying fractions $\frac{-\cos\left(x\right)}{\sin\left(x\right)} \times \frac{1}{\cos\left(x\right)}$

$\frac{1+\csc\left(x\right)+\frac{-\cos\left(x\right)}{\sin\left(x\right)\cos\left(x\right)}}{\sec\left(x\right)}$

Simplify the fraction $\frac{-\cos\left(x\right)}{\sin\left(x\right)\cos\left(x\right)}$ by $\cos\left(x\right)$

$\frac{1+\csc\left(x\right)+\frac{-1}{\sin\left(x\right)}}{\sec\left(x\right)}$

The reciprocal sine function is cosecant: $\frac{1}{\sin(x)}=\csc(x)$

$\frac{1+\csc\left(x\right)-\csc\left(x\right)}{\sec\left(x\right)}$
4

Simplify $-\cot\left(x\right)\sec\left(x\right)$ by applying trigonometric identities

$\frac{1+\csc\left(x\right)-\csc\left(x\right)}{\sec\left(x\right)}$
5

Cancel like terms $\csc\left(x\right)$ and $-\csc\left(x\right)$

$\frac{1}{\sec\left(x\right)}$
6

Applying the trigonometric identity: $\displaystyle\frac{1}{\sec(\theta)}=\cos(\theta)$

$\cos\left(x\right)$
7

Since we have reached the expression of our goal, we have proven the identity

true

Final answer to the problem

true

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